Bisect It

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-CO.C.9HSG-CO.D.12HSG-CO.A.3
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11 questions

Bisect It

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Why Does This Construction Work?

1) Explain what steps were taken to construct the perpendicular bisector in this image.

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2) Explain why these steps produce a line with the properties of a perpendicular bisector.

Construction from Definition

Han, Clare, and Andre thought of a way to construct an angle bisector. They used a circle to construct points DD and EE the same distance from AA. Then they connected DD and EE and found the midpoint of segment DEDE. They thought that ray AFAF would be the bisector of angle DAEDAE.

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3) Mark the given information of the diagram.

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

4) Han's rough-draft justification: FF is the midpoint of segment DEDE. I noticed that FF is also on the perpendicular bisection of angle DAEDAE.

5) Clare’s rough-draft justification: Since segment DADA is congruent to segment EAEA, triangle DEADEA is isosceles. DFDF has to be congruent to EFEF because they are the same length. So, AFAF has to be the angle bisector.

6) Andre’s rough-draft justification: What if you draw a segment from FF to AA? Segments DFDF and EFEF are congruent. Also, angle DAFDAF is congruent to angle EAFEAF. Then both triangles are congruent on either side of the angle bisector line.

7) Using your ideas about ways to make each student's explanation better, write your own explanation for why ray AFAF must be an angle bisector.

Reflecting on Reflection

8) Below is a diagram of an isosceles triangle APBAPB with segment APAP congruent to segment BPBP.

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

1. Segment APAP is congruent to segment BPBP because triangle APBAPB is isosceles.

2. The angle bisector of APBAPB intersects segment ABAB. Call that point QQ.

3. By the definition of angle bisector, angles APQAPQ and BPQBPQ are congruent.

4. Segment PQPQ is congruent to itself.

5. By the Side-Angle-Side Triangle Congruence Theorem, triangle APQAPQ must be congruent to triangle BPQBPQ.

6. Therefore the corresponding segments AQAQ and BQBQ are congruent and corresponding angles AQPAQP and BQPBQP are congruent.

7. Since angles AQPAQP and BQPBQP are both congruent and supplementary angles, each angle must be a right angle.

8. So PQPQ must be the perpendicular bisector of segment ABAB.

9. Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the triangle across PQPQ the vertex PP will stay in the same spot and the 2 endpoints of the base, AA and BB, will switch places.

10. Therefore the angle bisector PQPQ is a line of symmetry for triangle APBAPB.

Annotate the diagram with each piece of information in the proof.

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9) Write a summary of how this proof shows the angle bisector of a triangle is a line of symmetry.

10) Below is a diagram of parallelogram ABCDABCD.

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

1. The diagonals of a parallelogram intersect. Call that point MM.

2. The diagonals of a parallelogram bisect each other, so MBMB is congruent to MDMD.

3. By the definition of parallelogram, the opposite sides ABAB and CDCD are parallel.

4. Angles ABMABM and ADMADM are alternate interior angles of parallel lines so they must be congruent.

5. Segment AMAM is congruent to itself.

6. By the Side-Angle-Side Triangle Congruence Theorem, triangle ABMABM is congruent to triangle ADMADM.

7. Therefore the corresponding angles AMBAMB and AMDAMD are congruent.

8. Since angles AMBAMB and AMDAMD are both congruent and supplementary angles, each angle must be a right angle.

9. So ACAC must be the perpendicular bisector of segment BDBD.

10. Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the parallelogram across ACAC the vertices AA and CC will stay in the same spot and the 2 endpoints of the other diagonal, BB and DD, will switch places.

11. Therefore diagonal ACAC is a line of symmetry for parallelogram ABCDABCD.

Annotate the diagram with each piece of information in the proof.

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11) Find the errors that make this proof invalid, identifying any lines that have errors or false assumptions.