Mashup Math · YouTube · 9:29
This 160-second introductory geometry clip presents one worked two-column proof. After a brief branded intro, the screen shows a diagram with intersecting segments and meeting at F, the given that they bisect each other, and the goal GFK HFJ. The narrator explains the general proof setup, highlights the purple and pink target triangles, then fills the table row by row: first the given, next the congruent half-segments from the definition of bisect, then the congruent vertical angles at F, and finally the triangle congruence by SAS. Visual tick marks and angle arcs are synchronized with the algebraic statements, making the logical dependencies explicit. This 160-second clip works through a geometry two-column proof showing that ΔRSV ≅ ΔRUV. After introducing the problem structure, the narrator uses the given that R is the midpoint of SU to obtain SR ≅ UR, records the given SV ≅ UV, then adds the shared side VR ≅ VR by the Reflexive Property. With three pairs of congruent corresponding sides established, the proof concludes by the SSS postulate. This 160-second clip works through a geometry two-column proof whose goal is to show . After introducing the diagram and the givens and , the narrator explains that a segment-congruence target requires an extra step beyond triangle congruence. The proof first uses the congruent base angles to show is isosceles, then derives . Next it identifies vertical angles in the two lower triangles, applies ASA to obtain , and finishes with CPCTC to conclude . This clip completes a two-column geometry proof. The visible proof table shows the final steps: ΔBAE ≅ ΔCDE by AAS, then AE ≅ DE by CPCTC. The narrator explains that once the two yellow triangles are congruent, their corresponding parts are congruent, which finishes the original Prove statement. After the proof ends, the video switches to a promotional outro.
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Generated from the video's visuals and explanation; not verbatim speech.
The clip opens with a short Mashup Math logo animation and then cuts to a whiteboard-style geometry slide titled Practice With Proofs.
The narrator frames the task in general terms: every proof starts from a diagram, a given statement, and a conclusion to prove, organized in a statements-and-reasons table.
On the left, the diagram shows intersecting segments and meeting at F. The problem asks to prove , and the two relevant triangles are visually distinguished by purple and pink shading.
The first row of the table records the only given fact: and bisect each other at . Its reason is simply Given.
The second row converts that wording into geometric consequences. Because bisection means cutting into two equal halves, the proof states and , justified by Def. of Bisect. Matching tick marks appear on the diagram.
The third row notices the opposite angles at the intersection point. The proof writes and cites Vertical Angles, while green arcs mark the two angles on the figure.
With two side pairs and the included angle pair now established, the final row concludes by SAS. The narration emphasizes that the congruent angle lies between the two congruent sides, which is exactly the condition needed for the postulate.
The completed table and marked diagram together show the full logical chain from the given bisection statement to the triangle congruence conclusion.
The clip begins just after a previous example is finished. The narrator says that proof is done and announces a more challenging one, signaling that the lesson will continue in the same geometry-proof format.
A new problem appears with a diagram, two given facts, a statement to prove, and a blank two-column table labeled STATEMENTS and REASONS. The narrator explains that this proof has the same structure as the others: identify the diagram, use the givens, and prove two triangles congruent.
The target triangles are identified as RSV and RUV. On screen, RSV is shaded pink and RUV is shaded light green, making clear which two regions are being compared. The narrator also observes that together they form a larger right triangle.
The proof starts with the first given: R is the midpoint of . The narrator explains that a midpoint cuts a segment in half, so the immediate consequence is . This statement is entered in the table with the reason Def. of Midpoint, and matching tick marks are added to the diagram.
Since no further information follows from the first given, the narrator moves to the second given, . That statement is copied directly into the proof table with reason Given, and double tick marks are placed on the corresponding sides in the diagram.
After the explicit givens are used, the narrator turns to facts already known from geometry. The key observation is that the pink and light green triangles share side . Because it is the same segment in both triangles, the proof records with reason Reflexive Property.
At this point three pairs of corresponding congruent sides have been established: , , and . The final row therefore states RSV RUV with reason SSS, completing the two-column proof.
The clip opens by finishing the previous example with a reference to corresponding congruent sides, then announces one more practice proof.
A new geometry problem appears with the same two-column layout: a diagram on the left and a blank STATEMENTS/REASONS table on the right. The narrator says this proof is more challenging but structurally similar.
The key difference is the goal. Instead of proving two triangles congruent directly, the problem asks to prove two segments congruent, so the method will need an additional final step.
The target is . The diagram highlights in blue and in green so the viewer can see exactly which parts must eventually be matched.
The proof begins by entering the givens. First, is written in the table and marked on the diagram. Second, is entered and marked at vertices B and C.
With the givens used up, the narrator shifts to extracting new information from the figure. The congruent angles and are recognized as base angles of .
That observation triggers the Base Angle Theorem in the direction used here: if a triangle has two congruent base angles, then the triangle is isosceles. Therefore the third row states is isosceles.
From the isosceles-triangle definition, the proof derives the congruence of the sides opposite those angles, namely . Matching tick marks are added to the diagram.
The narrator pauses to explain proof strategy: even if it is not yet obvious how will help, it is worth recording justified intermediate facts because they often become the missing pieces later.
Attention then moves to the two lower triangles, and . At their shared vertex E, the proof identifies a pair of vertical angles, and , and uses the Vertical Angles theorem to conclude .
Now each lower triangle contains the ASA pattern: one angle congruence from the givens, the included side congruence derived from the isosceles triangle, and another angle congruence from vertical angles. Thus by ASA.
The final step applies CPCTC. Since the two triangles are congruent, their corresponding sides are congruent, giving the desired conclusion .
The clip opens on a finished geometry board: a colored diagram on the left and a two-column proof table on the right. The visible proof already contains the earlier statements ∠A ≅ ∠D (Given), ∠2 ≅ 3 (Given), ΔBCE is isosceles (Base Angle Theorem), BE ≅ CE (Def. of Isosceles Triangle), and ∠5 ≅ 6 (Vertical Angles).
At this point the narrator identifies the next logical move: the two yellow triangles, ΔBAE and ΔCDE, are congruent by AAS. The mathematical basis is that two corresponding angle pairs and one corresponding non-included side pair have already been established in the table.
The explanation then shifts from the whole-triangle congruence to the specific goal of the problem. Because AE and DE are parts of the now-congruent triangles, the proof can move toward the original Prove statement rather than stopping at triangle congruence alone.
The final row is completed with AE ≅ DE, justified by CPCTC. In other words, once ΔBAE ≅ ΔCDE has been proved, matching parts of those triangles inherit congruence, so the target segments are congruent.
The narrator restates the dependency in plain language: the yellow triangles were shown congruent in the second-to-last step, and that is what licenses the conclusion about their corresponding parts. This makes the proof order explicit: first triangle congruence, then segment congruence.
The lesson content ends here. The board remains on the completed proof briefly while the speaker gives closing remarks.
The video then cuts to a promotional outro with a presenter and channel links. No further geometry reasoning is added in this remaining interval.
The video presents a geometry proof as a table with two columns, STATEMENTS and REASONS, supported by a diagram, a given condition, and a target conclusion. This example uses that structure to move from one given fact to a triangle congruence result.
Given: and bisect each other at . Prove: . The purple and pink shaded triangles identify the two objects being compared.
When two segments bisect each other at a point, that point splits each segment into two congruent halves. In this problem, that gives and .
The opposite angles formed at the intersection point are vertical angles. The proof uses the fact that vertical angles are congruent to establish .
After obtaining two congruent side pairs and one congruent included angle pair, the proof applies the Side-Angle-Side postulate to conclude .
The full argument proceeds in four rows: 1) state the given bisection; 2) derive congruent half-segments by definition of bisect; 3) identify congruent vertical angles; 4) conclude triangle congruence by SAS.
This example is presented as a standard geometry two-column proof. The screen shows a diagram, the givens, the statement to prove, and a table with columns for statements and reasons. The narrator emphasizes that the structure is the same as earlier proofs: start from the diagram and givens, then build toward the congruence conclusion.
The problem asks to prove that triangle RSV is congruent to triangle RUV. In the diagram, these are the two smaller triangles inside a larger right triangle, distinguished visually by pink and light green shading.
The first given says that R is the midpoint of segment SU. By the definition of midpoint, R divides SU into two congruent parts, so the proof obtains SR congruent to UR. This becomes the first derived side congruence in the table.
The second given is used directly as a statement in the proof. It supplies another pair of congruent corresponding sides for the two triangles, and the diagram marks these sides with double ticks.
Both triangles contain the same side VR. Since a segment is congruent to itself, the proof adds VR congruent to VR and justifies it by the Reflexive Property. This step is important because it provides the third pair of congruent sides needed for SSS.
Once three pairs of corresponding sides are congruent, the triangles are congruent by the Side-Side-Side postulate. The completed proof uses SR ≅ UR, SV ≅ UV, and VR ≅ VR to conclude that triangle RSV is congruent to triangle RUV.
This example proves , not just a pair of congruent triangles. The video explains that the route is to prove the lower triangles congruent first and then extract the desired side congruence as a corresponding part.
The first two proof rows are the assumptions supplied by the problem: and . They are written as Given and also marked on the diagram with colored angle arcs.
Because and are congruent base angles of , the proof concludes that is isosceles. This is the first derived fact beyond the givens.
Once is known to be isosceles, the sides opposite the congruent angles are congruent. The proof records and later reuses this as the included side for the lower triangles.
In the two smaller triangles, angles 5 and 6 at point E are vertical angles. The proof uses the theorem that vertical angles are congruent to add .
The collected facts form an angle-side-angle pattern in and : , , and . Therefore the triangles are congruent by ASA.
After triangle congruence is established, the desired conclusion follows from corresponding parts of congruent triangles being congruent. This yields the final statement .
The narrator explicitly advises writing down justified consequences as soon as they are noticed, even if their role is not yet clear. In this proof, seems indirect at first but becomes essential for ASA.
The proof table uses AAS to conclude that the two yellow triangles are congruent. The visible ingredients are the angle congruences ∠A ≅ ∠D and 5 ≅ ∠6 together with the side congruence BE ≅ CE. This is the second-to-last statement in the worked proof.
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. After the triangles have already been proved congruent, matching parts such as AE and DE can be concluded congruent. The clip uses this principle to finish the original Prove statement.
The upper green triangle BCE is identified as isosceles, and the definition of an isosceles triangle gives BE ≅ CE. That side congruence is then one of the three pieces needed for the AAS argument about ΔBAE and ΔCDE.
Angles 5 and 6 are opposite angles formed at the intersection point E. The proof cites Vertical Angles to justify ∠5 ≅ 6, providing one of the two angle pairs used in the AAS step.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The given statement and the first table row use and .
The diagram shows two intersecting segments with endpoints G, J and H, K meeting at F.
,\
Line segments joining G to J and H to K; they intersect at F.
Geometric segments in the proof diagram.
The given says the segments bisect each other at F.
F is labeled at the intersection point of the two diagonals.
F
The common intersection point of and .
Point in the plane of the diagram.
The Prove line reads GFK HFJ.
The left triangle is shaded purple and the right triangle is shaded pink.
GFK,\ HFJ
The two triangles whose congruence is to be proved.
Triangles formed by the intersecting segments and outer sides.
Row 2 states \& .
Tick marks are added to the paired halves of the intersecting segments.
,
Congruent segment pairs obtained from mutual bisection at F.
Subsegments of and .
Row 3 states GFK HFJ.
Green angle arcs appear at F on opposite sides of the intersection.
GFK,\ HFJ
The pair of vertical angles at F used as the included angles in the two triangles.
Angles formed by intersecting segments at F.
The statement to prove is written as ΔRSV ≅ ΔRUV.
Triangle RSV is the pink triangle formed by vertices R, S, and V.
RSV
Triangle with vertices R, S, and V; in the diagram it is the pink triangle.
Geometric figure in the proof diagram
The statement to prove is written as ΔRSV ≅ ΔRUV.
Triangle RUV is the light green triangle formed by vertices R, U, and V.
RUV
Triangle with vertices R, U, and V; in the diagram it is the light green triangle.
Geometric figure in the proof diagram
Given text states: R is the midpoint of .
Segment SU is the hypotenuse-like side of the larger right triangle, with R on it.
Line segment with endpoints S and U, containing point R.
Segment in the diagram
Given text states: R is the midpoint of .
Point R lies on segment SU between S and U.
R
Point on segment SU identified as its midpoint.
Point in the diagram
The proof table writes .
Tick marks appear on SR and UR after the midpoint step.
Subsegment of from S to the midpoint R.
Segment in the diagram
The proof table writes .
Tick marks appear on SR and UR after the midpoint step.
Subsegment of from the midpoint R to U.
Segment in the diagram
Given text states .
Double tick marks are shown on SV and UV.
Side shared by the large triangle and triangle RSV.
Segment in the diagram
The narrator says proofs have a diagram, given information, something to prove, and a statement-and-reason table.
The screen shows a geometry problem layout with Given, Prove, and a two-column table labeled STATEMENTS and REASONS.
The video introduces a geometry proof as a logical argument built from a diagram, a given statement, a target conclusion, and a table pairing each statement with its reason.
Used for the worked example in this clip.
The narrator explains that when segments bisect each other, they cut each other into two equal halves.
The second row reason is written as Def. of Bisect.
If two segments bisect each other at a point, that point divides each segment into two congruent parts. In this problem, that gives and .
Applies when the given states mutual bisection at F.
The narrator says all vertical angles are congruent.
Row 3 states GFK HFJ with reason Vertical Angles.
Opposite angles at F are marked with matching green arcs.
The proof uses the standard fact that opposite angles formed by intersecting lines or segments are congruent, here applied to GFK and HFJ.
Requires two intersecting segments or lines forming opposite angles at the intersection.
The narrator says there are two congruent sides and one congruent angle in between them, so the side-angle-side postulate can be used.
The final row reason is SAS.
The diagram already shows two marked side pairs and one marked angle pair before the final conclusion.
The clip applies SAS by combining two pairs of congruent sides with the congruent included angle to conclude triangle congruence.
The congruent angle must lie between the two congruent sides in each triangle.
The screen shows a diagram, Given/Prove text, and a blank table headed STATEMENTS and REASONS.
The narrator says the proof has the same structure as earlier ones: a diagram, two pieces of given information, and a goal to prove two triangles congruent.
The video presents a geometry proof in a standard two-column format. The left side lists statements, the right side lists reasons, and the problem supplies a diagram plus given facts and a target conclusion.
A diagram is provided.
Given information is listed separately from the statement to prove.
The proof is organized as statements with matching reasons.
The narrator says a midpoint cuts a segment in half.
The table records with reason Def. of Midpoint.
Tick marks are added to SR and UR after the midpoint statement.
If a point is the midpoint of a segment, then it divides that segment into two congruent subsegments. In this problem, because R is the midpoint of , the proof concludes .
The point must lie on the segment.
The conclusion is congruence of the two subsegments determined by the midpoint.
The narrator explains that both triangles share side VR, so VR has the same length in each triangle.
The table records with reason Reflexive Property.
Segment VR is visually emphasized as the common side between the pink and light green triangles.
When two triangles share the same side, that side is congruent to itself. Here the shared side supplies the third pair of congruent sides needed for triangle congruence.
The same segment must belong to both triangles being compared.
This step is used after the explicitly given congruences have been recorded.
The narrator says the triangles are congruent by using the side-side-side postulate and that each triangle shares three pairs of congruent sides.
The final row states RSV RUV with reason SSS.
Three pairs of corresponding sides are marked before the final conclusion.
If three pairs of corresponding sides of two triangles are congruent, then the triangles are congruent. The proof first establishes , , and , then applies SSS to conclude RSV RUV.
Three corresponding side pairs must be congruent.
The correspondence of vertices must match the order in the triangle names.
The narrator says this proof has the same structure: diagram, givens, and prove statement.
A geometry figure appears on the left and a blank two-column table labeled STATEMENTS and REASONS appears on the right.
The problem text lists Given: , ; Prove: .
The video introduces a new practice proof with the standard layout: a diagram, a list of givens, a prove statement, and a two-column table. The method emphasized here is that when the goal is to prove two segments congruent rather than two triangles congruent, the proof needs an extra stage: first establish triangle congruence, then extract the desired segment congruence.
The proof uses a diagram plus givens.
The final target is a pair of congruent segments, not directly a pair of congruent triangles.
First table row: with reason Given.
The narrator says, use our first given, that angle A and angle D are congruent.
Orange arcs appear at A and D.
The first supplied fact is entered directly into the proof table as a given. It marks one angle in each of the two lower triangles and becomes part of the eventual ASA pattern.
This is an assumption stated in the problem, not derived within the clip.
Second table row: with reason Given.
The narrator says the second given tells us angle 2 and angle 3 are congruent.
Purple arcs appear at B and C.
The second supplied fact is also entered directly as a given. These are the two base angles of the upper triangle , and the proof immediately uses them to infer that the upper triangle is isosceles.
This is an assumption stated in the problem, not derived within the clip.
The narrator says triangle BCE has two congruent base angles, so it is isosceles because of the base angle theorem.
Third table row: is isosceles with reason Base Angle Theorem.
Triangle BCE is shaded to identify the triangle under discussion.
The video uses the converse direction of the base-angle idea: if a triangle has two congruent base angles, then the triangle is isosceles. Here are identified as the base angles of , so the triangle is labeled isosceles in the proof table.
The two congruent angles must be the base angles of the same triangle.
In this clip, the relevant triangle is .
The narrator explicitly says all vertical angles are congruent.
The table records GFK HFJ with reason Vertical Angles.
If two angles are vertical angles, then they are congruent. In this example, GFK HFJ.
The angles are formed by intersecting segments at F.
They are opposite (vertical) angles.
For the specific pair of vertical angles shown in the diagram.
The narrator says two congruent sides and one congruent angle in between them justify using the side-angle-side postulate.
The final row concludes GFK HFJ with reason SAS.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Here this yields GFK HFJ.
GFK HFJ
The congruent angle is the included angle between the two congruent sides.
Applied to the two triangles named in the proof.
The narrator states that a midpoint cuts a segment in half.
The proof table writes with reason Def. of Midpoint.
If R is the midpoint of , then .
R is the midpoint of .
For the specific points R, S, and U in the diagram.
The narrator says whatever the length of VR is in one triangle, it has to be the same length in the other triangle.
The proof table writes with reason Reflexive Property.
Because is shared by RSV and RUV, .
The two triangles share side .
For the segment appearing in both triangles.
The narrator says the triangles are congruent by using the side-side-side postulate.
The final row states RSV RUV with reason SSS.
If , , and , then RSV RUV by SSS.
For triangles RSV and RUV with the stated correspondence.
The narrator states that because triangle BCE has two congruent base angles, it is isosceles.
Table row: is isosceles, reason Base Angle Theorem.
In , from the video concludes is isosceles.
and are angles of the same triangle .
is given.
For the specific triangle in the diagram.
The narrator says all vertical angles are congruent.
Table row: , reason Vertical Angles.
Because and are vertical angles, .
and are formed as vertical angles at the intersection point E.
Universal statement invoked for this particular pair of vertical angles.
The narrator says the two yellow triangles have two consecutive congruent angles followed by a congruent side.
Table row: , reason ASA.
From , , and , the video concludes by ASA.
The three named correspondences hold in the two triangles.
The side congruence lies between the two angle congruences in the ASA order.
Applied to the specific pair and .
The narrator explicitly names CPCTC as corresponding parts of congruent triangles are congruent.
Final table row: , reason CPCTC.
Since , their corresponding sides satisfy .
has already been proved.
and are corresponding sides under that congruence.
Applied to the specific corresponding pair in the proved triangle congruence.
The proof table states ΔBAE ≅ ΔCDE with reason AAS.
The narrator explicitly says the two triangles are congruent by the angle-angle-side postulate.
From the displayed proof, ΔBAE ≅ ΔCDE follows by the AAS postulate using ∠A ≅ ∠D, ∠5 ≅ 6, and BE CE.
A ≅ ∠D is given.
∠5 ≅ ∠6 because they are vertical angles.
BE ≅ CE because ΔBCE is isosceles.
For the specific triangles ΔBAE and ΔCDE in the shown diagram.
The last row of the proof table states AE ≅ DE with reason CPCTC.
The narrator says AE is congruent to DE because corresponding parts of congruent triangles are congruent.
After proving ΔBAE ≅ ΔCDE, the corresponding segments AE and DE are congruent.
ΔBAE ≅ ΔCDE has already been proved.
AE and DE are corresponding parts under that triangle congruence.
For the specific corresponding segments AE and DE in the proved triangle congruence.
The table is filled row by row: Given; Def. of Bisect; Vertical Angles; SAS.
The narrator explains each step in the same order as the rows appear.
Tick marks and angle arcs are added as corresponding statements are introduced.
Start from the single given statement in the problem.
Given
Mutual bisection at F means each segment is split into two congruent halves.
Definition of bisect
The opposite angles at the intersection point F are vertical angles, so they are congruent.
Vertical angles theorem
Each triangle now has two congruent sides and the included angle congruent, so the triangles are congruent by SAS.
SAS postulate
The proof establishes GFK HFJ using the given bisection, the definition of bisect, vertical angles, and SAS.
The proof table is filled row by row with statements and reasons.
The narrator explains each step in order: use the givens, identify the shared side, then apply SSS.
Corresponding side marks are added as each congruence is established.
Record the first given fact in the statements column.
Given
Convert the midpoint information into a congruence of the two subsegments of .
Def. of Midpoint
Record the second given fact directly as a pair of congruent sides.
Given
Identify the side shared by both triangles and state that it is congruent to itself.
Reflexive Property
Use the three established pairs of congruent corresponding sides to conclude the triangles are congruent.
SSS
The proof establishes RSV RUV by first obtaining three pairs of congruent corresponding sides and then applying the SSS postulate.
The completed two-column table lists seven rows from the givens through CPCTC.
The narrator explains each row in sequence and names the reasons Given, Base Angle Theorem, Def. of Isosceles Triangle, Vertical Angles, ASA, and CPCTC.
Angles and sides are progressively marked as each statement is added.
Enter the first given into the proof table.
Given.
Enter the second given into the proof table.
Given.
Use the congruent base angles of the upper triangle to classify it as isosceles.
Base Angle Theorem.
From the isosceles triangle, conclude that the sides opposite the congruent angles are congruent.
Def. of Isosceles Triangle.
Identify the opposite angles at E in the two lower triangles as vertical angles.
Vertical Angles.
Combine the two angle congruences and the included side congruence in each lower triangle.
ASA.
Extract the desired segment congruence as a pair of corresponding parts of the congruent triangles.
CPCTC.
The proof establishes the required result by first proving and then applying CPCTC.
The narrator says you may not know whether BE congruent CE will help, but you should label as much as you can because more information makes proofs easier.
After deriving , the narrator acknowledges that its usefulness may not yet be obvious.
Pedagogical commentary during proof construction.
The advice is to keep labeling available consequences on the diagram and in the table, even before seeing exactly how each fact will be used.
Heuristic for proof planning.
The clip presents proof building as partly exploratory: record justified intermediate facts, then look for how they combine into a congruence criterion.
The proof table visibly contains the ordered statements ∠A ≅ ∠D, ∠2 ≅ 3, ΔBCE is isosceles, BE ≅ CE, ∠5 ≅ ∠6, ΔBAE ≅ ΔCDE, and AE ≅ DE with their reasons.
The narrator explains the AAS step and then the CPCTC step that finishes the proof.
The clip begins with the proof already partly filled in, so the exact earlier setup before 0 seconds is not available in this excerpt.
The first listed statement is the given congruence of angle A and angle D.
Given
The second listed statement is the given congruence of angle 2 and angle 3.
Given
The proof then identifies the upper green triangle as isosceles.
Base Angle Theorem
From the isosceles-triangle identification, the two equal sides are concluded congruent.
Def. of Isosceles Triangle
The angles at point E opposite one another are declared congruent.
Vertical Angles
With two angle pairs and a corresponding non-included side pair established, the two yellow triangles are proved congruent.
AAS
Once the triangles are congruent, the target corresponding segments are concluded congruent, completing the proof.
CPCTC
The displayed proof ends with AE ≅ DE, which matches the original Prove statement.
The slide title is Practice With Proofs, with Given: and bisect each other at F. Prove: GFK HFJ.
The narrator works through this as the first example of the lesson.
A quadrilateral-like figure with intersecting diagonals is shown, and the two side triangles are highlighted.
The outer figure resembles a square or rectangle visually, but the video does not state that it is a square or rectangle, so no such property should be assumed.
Given that and bisect each other at F, prove that GFK HFJ.
and bisect each other at F.
Prove GFK HFJ.
Record the given information as the first statement in the proof table.
Given
Translate the bisection condition into congruent half-segments.
Def. of Bisect
Identify the opposite angles at F as vertical angles and mark them congruent.
Vertical Angles
Use the two side pairs and the included angle pair to conclude triangle congruence.
SAS
GFK HFJ
The completed table matches the visual markings on the diagram: two congruent side pairs and one congruent included angle pair.
The new problem shows a large right triangle with point R on hypotenuse and segment drawn inside.
Given: R is the midpoint of ; . Prove: RSV RUV.
The narrator introduces this as a more challenging proof and works through it completely.
Given that R is the midpoint of and , prove that RSV RUV.
R is the midpoint of .
.
Prove RSV RUV.
Start with the first given statement.
Given
From the midpoint condition, the two parts of are congruent.
Def. of Midpoint
Use the second given statement directly.
Given
Notice that is the common side of the two triangles.
Reflexive Property
With three pairs of congruent corresponding sides established, conclude triangle congruence.
SSS
RSV RUV
The completed two-column table lists exactly the three side congruences required for SSS, and the diagram marks those corresponding sides before the final conclusion.
Problem text: Given , ; Prove .
The narrator calls this one more challenging practice proof and works through it completely.
The figure shows intersecting segments forming , , and .
Given and , prove .
Diagram with triangles , , and meeting at E
Prove the two lower-triangle sides and are congruent.
Write the first given in the statements column.
Given.
Write the second given in the statements column.
Given.
Recognize and as congruent base angles of the upper triangle.
Base Angle Theorem.
Convert the isosceles-triangle conclusion into congruence of the opposite sides.
Def. of Isosceles Triangle.
Mark the opposite angles at E in the two lower triangles.
Vertical Angles.
Use the two angle congruences and the included side congruence to prove the lower triangles congruent.
ASA.
Take the desired segments as corresponding parts of the congruent triangles.
CPCTC.
The final table row matches the original Prove statement, and each prior row is justified by a named theorem, definition, or given.
The screen shows a geometry diagram with points A, B, C, D, E and a two-column proof table.
The given/prove area reads Given: ∠A ≅ ∠D, 2 ≅ ∠3; Prove: AE ≅ DE.
The narrator walks through the last two proof steps and states the final conclusion.
Only the ending portion of the worked proof is present in this clip; earlier instructional setup is outside the excerpt.
Given the displayed figure with Given: ∠A ≅ ∠D and 2 ≅ ∠3, prove that AE ≅ DE.
∠A ≅ ∠D
2 ≅ ∠3
Diagram with triangles BAE, CDE, and BCE meeting at point E
Prove .
The proof uses the base-angle information to identify triangle BCE as isosceles.
Base Angle Theorem
Because triangle BCE is isosceles, its equal sides are congruent.
Def. of Isosceles Triangle
The opposite angles at intersection point E are congruent.
Vertical Angles
Using the two angle pairs and the side pair just established, the two yellow triangles are congruent.
AAS
The desired segments are corresponding parts of the congruent triangles, so they are congruent.
CPCTC
The final statement in the proof table exactly matches the Prove line shown at the lower left of the screen.
The left triangle becomes purple and the right triangle becomes pink.
The narrator refers to the purple triangle and the pink triangle while explaining the goal.
GFK
HFJ
The left triangle is filled purple.
The right triangle is filled pink.
The underlying diagram and labels remain unchanged.
The color coding identifies exactly which two triangles the proof must show congruent.
The intersecting segments are emphasized and tick marks are added to paired halves.
The second row states \& .
The exact tick style varies slightly across frames, but the paired equality relation is clear.
Matching tick marks are added to the two halves of each intersecting segment.
The intersection point remains F.
The overall quadrilateral outline stays fixed.
The visual marks encode the consequence of the given bisection: each segment is split into congruent parts.
Green arcs appear on opposite angles at F.
The third row states GFK HFJ.
GFK
HFJ
Opposite angles at F receive matching green arc marks.
The previously added segment marks remain visible.
The arcs identify the included angle pair needed for SAS.
The STATEMENTS/REASONS table fills in one row at a time until four rows are present.
Final visible rows are Given; Def. of Bisect; Vertical Angles; SAS.
Statements column
Reasons column
Row 1 appears with the given statement.
Row 2 adds the congruent segment pairs.
Row 3 adds the congruent vertical angles.
Row 4 adds the triangle congruence conclusion with SAS.
The diagram remains on the left while the table is built on the right.
The animation demonstrates how a formal proof is assembled step by step from givens to conclusion.
The previous completed proof disappears and a new diagram with a blank STATEMENTS/REASONS table appears.
Previous completed proof table
New triangle diagram
Blank two-column table
The screen switches from a finished example to a new unsolved problem.
A blank proof table replaces the previously filled table.
The overall lesson format remains a geometry proof with diagram plus two-column table.
This visual transition marks the start of a harder practice problem after the narrator says they will move on to a more challenging one.
Triangle RSV is filled pink and triangle RUV is filled light green.
The narrator identifies the two triangles to be proved congruent and notes that together they form a larger right triangle.
Pink triangle RSV
Light green triangle RUV
Larger right triangle SUV
The two smaller triangles are colored differently to distinguish them.
Their union is visually recognized as the larger triangle.
Vertex labels R, S, U, and V remain fixed.
The shared side VR belongs to both colored triangles.
The coloring clarifies which two triangles are the objects of the congruence proof and how they sit inside the larger figure.
Single tick marks appear on and after the midpoint statement is entered.
The table shows with reason Def. of Midpoint.
Segment
Segment
Point R
Tick marks are added to SR and UR.
The proof table gains the congruence statement derived from the midpoint.
R remains on segment SU.
The larger diagram geometry does not change.
The visual tick marks encode the mathematical consequence that the midpoint divides into two congruent pieces.
Double tick marks appear on and .
The table records with reason Given.
Segment
Segment
Double tick marks are added to SV and UV.
The second given is transferred into the proof table.
The side labels and triangle positions remain unchanged.
The double tick marks distinguish this given pair of congruent sides from the earlier midpoint-derived pair.
Segment is highlighted as the shared side of the pink and light green triangles.
The table records with reason Reflexive Property.
Segment
Triangle RSV
Triangle RUV
The common side is visually singled out.
The reflexive congruence statement is added to the table.
VR is the same segment in both triangles.
No new external information is introduced; the step comes from the diagram structure.
The highlight makes explicit the geometric observation that the two triangles share a side, which justifies using the reflexive property.
The final row of the table fills in with RSV RUV and reason SSS.
All three pairs of corresponding sides are already marked before the conclusion.
Completed two-column table
Marked side pairs
Triangles RSV and RUV
The conclusion statement appears in the last row.
The reason SSS is written beside it.
The three side congruences established earlier remain the basis for the conclusion.
The visual completion of the table shows that once three corresponding side pairs are congruent, the triangle congruence follows immediately by SSS.
The previous completed proof disappears and a new diagram with a blank two-column table appears.
The narrator says they are done with this proof and have one more.
previous completed proof table
new geometry diagram
blank STATEMENTS/REASONS table
Old proof is removed.
New problem with Given and Prove text appears.
Table resets to empty rows.
The lesson continues in the same two-column proof format.
The visual reset signals a new worked example rather than a continuation of the previous argument.
Segment AE turns blue and segment DE turns green while the Prove line is emphasized.
The narrator says they will label AE in blue and DE in green.
segment AE
segment DE
Prove statement
AE is highlighted blue.
DE is highlighted green.
The underlying diagram geometry does not change.
The color coding isolates the exact conclusion to be proved before the intermediate triangle work begins.
The outer figure looks like a square or rectangle, but the only stated given is that and bisect each other at F.
Because the diagram resembles a square or rectangle, one might incorrectly use right angles, equal outer sides, or diagonal perpendicularity.
Only the explicitly given bisection information is used in the proof; no additional quadrilateral properties are stated in this clip.
The narrator specifically says the congruent angle is in between the two congruent sides.
The proof uses GFK and HFJ together with the adjacent side pairs.
Any congruent angle plus two congruent sides would prove triangle congruence.
The video's reasoning depends on the angle being between the two marked sides; that is why SAS applies here.
The narrator says that once the givens are all used up, we have to apply what we already know about geometry.
The first two rows come from the givens, and only afterward is the shared side identified.
One might try to jump straight to a congruence postulate without first translating the given statements into usable side congruences.
The video's method is to record and transform the givens first, then add geometric facts such as the shared side, and only then apply SSS.
The narrator explicitly points out that both triangles share side VR.
The proof includes a separate row with reason Reflexive Property.
Because the shared side is obvious in the diagram, one might think it does not need to be written in the proof.
The video treats the common side as a formal step: it must be stated as and justified by the Reflexive Property.
The narrator says unlike the last examples, this one asks to prove two line segments congruent, not just two triangles, so there is an extra step.
Assuming the proof is complete once two triangles are shown congruent.
The video explicitly states that when the target is , triangle congruence is only an intermediate result; one more step, CPCTC, is needed.
The narrator says you may not be sure whether BE congruent CE will help, but you should label as much as you can because more information makes proofs easier.
Believing a derived statement is useless just because its role is not immediately visible.
The clip advises recording justified intermediate conclusions first; later becomes the side needed for ASA.
The narrator emphasizes that AE and DE are congruent because the triangles are already congruent.
CPCTC appears only after the row ΔBAE ≅ ΔCDE.
A learner might try to conclude AE ≅ DE directly from scattered angle facts without first proving the whole triangles congruent.
In the displayed proof, CPCTC is valid only after ΔBAE ≅ ΔCDE has been established by AAS. The segment congruence is a consequence of the triangle congruence, not a substitute for proving it.
The narrator derives congruent half-segments directly from the phrase bisect each other.
The reason Def. of Bisect is attached to the segment congruence row.
The definition of bisect supplies the two congruent side pairs used inside the proof.
Row 3 records GFK HFJ with reason Vertical Angles.
The vertical angles fact provides the included angle congruence needed before the final SAS step.
The narrator checks whether enough information is available and then cites two sides and the angle between them.
The final row uses SAS after the previous three rows have been established.
The SAS conclusion depends on the earlier segment congruences and vertical-angle congruence already proved in the table.
The narrator describes the general proof structure before working the example.
The example is carried out in a STATEMENTS/REASONS table.
The worked example is presented using the two-column proof format introduced at the start.
The midpoint step produces , which later appears as one of the three side pairs used for SSS.
The definition of midpoint supplies one of the three required pairs of congruent corresponding sides for the SSS conclusion.
The reflexive step produces , which is then used in the final SSS row.
The shared-side congruence obtained from the reflexive property provides the third pair of congruent sides needed to apply SSS.
The entire example is carried out inside a STATEMENTS/REASONS table.
The general two-column proof format is the organizing method used to present the worked example.
Row 2 gives and row 3 concludes is isosceles.
The narrator directly connects the congruent base angles to the isosceles conclusion.
The second given is the premise used to invoke the Base Angle Theorem for .
Row 3 states is isosceles and row 4 states .
The narrator says the sides opposite the congruent angles are congruent because that is the definition of an isosceles triangle.
The classification of as isosceles is what licenses the deduction .
Row 4 provides and row 6 applies ASA to and .
The narrator later says the yellow triangles have two consecutive congruent angles followed by a congruent side.
The derived side congruence becomes the included side needed for the ASA argument in the lower triangles.
Row 5 gives and row 6 concludes by ASA.
The narrator explicitly uses the vertical-angle pair as part of the angle-side-angle pattern.
The vertical-angle congruence supplies the second angle needed for ASA.
Row 6 states and row 7 states by CPCTC.
The narrator says the last step is CPCTC.
Triangle congruence is the premise that allows extraction of the corresponding side congruence.
The narrator explains that a proof begins from the diagram, given information, and what must be proved.
The first table row is the given statement with reason Given.
The narrator says bisecting segments cut each other into two equal halves.
The row states \& with reason Def. of Bisect.
The narrator identifies the opposite angles at F as vertical angles and says they are congruent.
Matching green arcs mark GFK and HFJ.
The narrator says there are two congruent sides and one congruent angle in between them.
The last row concludes GFK HFJ with reason SAS.
The table shows with reason Def. of Midpoint.
The table shows with reason Reflexive Property.
The final reason written in the table is SSS.
The rows appear in the order: given midpoint, midpoint consequence, given side congruence, reflexive side, SSS conclusion.
Segment VR is highlighted as the common side of the pink and light green triangles.
The narrator says this proof has an extra step because the goal is two line segments, not just two triangles.
The narrator says triangle BCE is isosceles because of the base angle theorem.
Table row: is isosceles, reason Base Angle Theorem.
The narrator first derives BE congruent CE, then later uses it in the ASA pattern for the yellow triangles.
Rows 4 and 6 connect to .
Covered · Intro logo animation with no mathematical content.
Covered · Title slide and spoken introduction to the lesson and its purpose.
Covered · Explanation of proof structure and identification of the two triangles to compare.
Covered · First two proof rows: given statement and congruent half-segments from bisection.
Covered · Third proof row: vertical angles at F are congruent.
Covered · Final proof row and conclusion by SAS.
Covered · Trailing end of the clip after the proof is complete; no new mathematical content.
Covered · The clip opens on the tail end of a previous completed proof while the narrator says that example is done and introduces a more challenging one; this functions as transition context for the new proof format.
Covered · The new problem is displayed with diagram, givens, prove statement, and blank two-column table; the narrator describes the structure of the proof.
Covered · The narrator identifies the two triangles to compare, and the diagram colors triangle RSV pink and triangle RUV light green.
Covered · The first given is entered, then converted into by the definition of midpoint, with matching tick marks added.
Covered · The narrator checks off the first given and records the second given , adding double tick marks to the diagram.
Covered · After the givens are used, the narrator identifies the shared side and writes by the Reflexive Property.
Covered · With three pairs of congruent corresponding sides established, the final row states RSV RUV with reason SSS.
Covered · Opening seconds finish the previous example with the phrase about corresponding congruent sides and announce one more proof; no new mathematical content from the next problem is developed yet.
Covered · New problem is introduced with diagram, two givens, and a segment-congruence goal requiring an extra step.
Covered · Target segments AE and DE are identified and color-coded on the diagram.
Covered · The two given angle congruences are entered into the proof table and marked on the figure.
Covered · The narrator shifts from using givens to extracting more information and proves is isosceles.
Covered · From the isosceles triangle, the proof derives .
Covered · Strategic commentary explains why it is useful to record intermediate facts even before seeing their final role.
Covered · The two lower triangles are identified and their vertical angles at E are proved congruent.
Covered · ASA is applied to prove , then CPCTC yields .
Covered · The narrator states that ΔBAE and ΔCDE are congruent by AAS while the proof table shows the corresponding row and reason.
Covered · The speaker transitions from the triangle-congruence result to the target segments AE and DE, explaining that they are parts of the congruent triangles.
Covered · The final proof step is completed: AE ≅ DE is justified by CPCTC, and the narrator closes the lesson.
Covered · Promotional outro with presenter and channel links; no additional mathematical content for the triangle-congruence proof.
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