Han, Clare, and Andre were given the following task: “Construct an angle bisector. Write a proof that the ray you constructed is the angle bisector of angle .”

Read the script your teacher will give you. After each sentence, decide if there is anything to add to the diagram.

With your group, discuss each student’s approach. For each approach, answer these questions:

• What do you notice that this student understands about the problem?
• What question would you ask them to help them move forward?
   

Construct an angle bisector. Write a proof that the ray you constructed is the angle bisector of angle . 

1. Here is a diagram of an isosceles triangle  with segment congruent to segment .

   

   Here is a valid proof that the angle bisector of the vertex angle of an isosceles triangle is a line of symmetry.

   1. Read the proof, and annotate the diagram with each piece of information in the proof.
   2. 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 .

2. Here is a diagram of parallelogram .​​​

   

   Here is an invalid proof that a diagonal of a parallelogram is a line of symmetry.

   1. Read the proof, and annotate the diagram with each piece of information in the proof.
   2. Find the errors that make this proof invalid. Highlight any lines that have errors or false assumptions.
   • The diagonals of a parallelogram intersect. Call that point .
   • The diagonals of a parallelogram bisect each other, so is congruent to .
   • By the definition of parallelogram, the opposite sides and are parallel.
   • Angles and are alternate interior angles of parallel lines, so they must be congruent.
   • Segment is congruent to itself.
   • By the Side-Angle-Side Triangle Congruence Theorem, triangle is congruent to triangle .
   • Therefore, the 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 parallelogram across , the vertices  and  will stay in the same spot, and the 2 endpoints of the other diagonal, and , will switch places.
   • Therefore, diagonal  is a line of symmetry for parallelogram .