If you are Partner A, explain to your partner what steps were taken to construct the perpendicular bisector in this image.[br][br]If you are Partner B, listen to your partner’s explanation, and then explain to your partner why these steps produce a line with the properties of a perpendicular bisector.[br][br]Then, work together to make sure the main steps in Partner A’s explanation have a reason from Partner B’s explanation.

[size=150]They used a circle to construct points [math]D[/math] and [math]E[/math] the same distance from [math]A[/math]. Then they connected [math]D[/math] and[math]E[/math] and found the midpoint of segment [math]DE[/math]. They thought that ray [math]AF[/math] would be the bisector of angle [math]DAE[/math]. Mark the given information on the diagram:[/size][br][br]Han’s rough-draft justification: [math]F[/math] is the midpoint of segment [math]DE[/math]. I noticed that [math]F[/math] is also on the perpendicular bisector of angle [math]DAE[/math].[br][br]Clare’s rough-draft justification: Since segment [math]DA[/math] is congruent to segment [math]EA[/math], triangle [math]DEA[/math] is isosceles. [math]DF[/math] has to be congruent to [math]EF[/math] because they are the same length. So, [math]AF[/math] has to be the angle bisector.[br][br]Andre’s rough-draft justification: What if you draw a segment from [math]A[/math] to [math]F[/math]? Segments [math]DF[/math] and [math]EF[/math] are congruent. Also, angle [math]DAF[/math] is congruent to angle [math]EAF[/math]. Then both triangles are congruent on either side of the angle bisector line.[br][br][size=150]Each student tried to justify why their construction worked. With your partner, discuss each student’s approach.[/size][list][*]What do you notice that this student understands about the problem?[/*][/list]

[list][*]What question would you ask them to help them move forward?[/*][/list]

Using the ideas you heard and the ways that each student could make their explanation better, write your own explanation for why ray [math]AF[/math] must be an angle bisector.

[size=150]Read the proof and annotate the diagram with each piece of information in the proof.[br][br]Write a summary of how this proof shows the angle bisector of the vertex angle of an isosceles triangle is a line of symmetry.[/size][br][br][list][*]Segment [math]AP[/math] is congruent to segment [math]BP[/math] because triangle [math]APB[/math] is isosceles.[/*][*]The angle bisector of [math]APB[/math] intersects segment [math]AB[/math]. Call that point [math]Q[/math].[/*][*]By the definition of angle bisector, angles [math]APQ[/math] and [math]BPQ[/math] are congruent. [/*][*]Segment [math]PQ[/math] is congruent to itself.[/*][*]By the Side-Angle-Side Triangle Congruence Theorem, triangle [math]APQ[/math] must be congruent to triangle [math]BPQ[/math].[/*][*]Therefore the corresponding segments [math]AQ[/math] and [math]BQ[/math] are congruent and corresponding angles [math]AQP[/math] and [math]BQP[/math] are congruent.[/*][*]Since angles [math]AQP[/math] and [math]BQP[/math] are both congruent and supplementary angles, each angle must be a right angle.[/*][*]So [math]PQ[/math] must be the perpendicular bisector of segment [math]AB[/math].[/*][*]Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the triangle across [math]PQ[/math] the vertex [math]P[/math] will stay in the same spot and the 2 endpoints of the base, [math]A[/math] and [math]B[/math], will switch places.[/*][*]Therefore the angle bisector [math]PQ[/math] is a line of symmetry for triangle [math]APB[/math].[/*][/list]

What is an example of such a quadrilateral?[br]

How would you modify this proof to be a valid proof for that type of quadrilateral?[br]