IM Geo.2.14 Lesson: Bisect It

If you are Partner A, explain to your partner what steps were taken to construct the perpendicular bisector in this image. 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. Then, work together to make sure the main steps in Partner A’s explanation have a reason from Partner B’s explanation.

They used a circle to construct points  and  the same distance from . Then they connected  and and found the midpoint of segment . They thought that ray  would be the bisector of angle . Mark the given information on the diagram: Han’s rough-draft justification:  is the midpoint of segment . I noticed that  is also on the perpendicular bisector of angle . Clare’s rough-draft justification: Since segment  is congruent to segment , triangle  is isosceles.  has to be congruent to  because they are the same length. So,  has to be the angle bisector. Andre’s rough-draft justification: What if you draw a segment from  to ? Segments  and  are congruent. Also, angle  is congruent to angle . Then both triangles are congruent on either side of the angle bisector line. Each student tried to justify why their construction worked. With your partner, discuss each student’s approach.

• What do you notice that this student understands about the problem?

• What question would you ask them to help them move forward?

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

Read the proof and annotate the diagram with each piece of information in the proof. Write a summary of how this proof shows the angle bisector of the vertex angle of an isosceles triangle is a line of symmetry.

• Segment  is congruent to segment  because triangle  is isosceles.
• The angle bisector of  intersects segment . Call that point .
• By the definition of angle bisector, angles  and  are congruent. 
• Segment  is congruent to itself.
• By the Side-Angle-Side Triangle Congruence Theorem, triangle  must be congruent to triangle .
• Therefore the corresponding segments  and  are congruent and corresponding angles  and  are congruent.
• Since angles  and  are both congruent and supplementary angles, each angle must be a right angle.
• So  must be the perpendicular bisector of segment .
• Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the triangle across  the vertex  will stay in the same spot and the 2 endpoints of the base,  and , will switch places.
• Therefore the angle bisector  is a line of symmetry for triangle .

What is an example of such a quadrilateral?

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