14.1: Why Does This Construction Work? (5 minutes)
In a previous unit, students conjectured that drawing circles with the same radius centered at the endpoints of a segment will construct its perpendicular bisector. In this activity, they return to that construction and now are able to justify that the steps work.
Monitor for students who:
- understand what it means to create the intersection of same-size circles (finding points that are an equal distance from the two centers)
- understand that two points are enough to define a line
Arrange students in groups of 2. Designate one student as Partner A and one student as Partner B.
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.
If students struggle to understand the given image, suggest they re-create the construction on their own.
Select students to share who can explain each of the two key points:
- the intersection of same-size circles gives points that are an equal distance from the two centers (which are the endpoints)
- two points define a unique line
14.2: Construction from Definition (25 minutes)
Students read three rough drafts of proofs and then work to write their own proof. Each rough draft provides support on a different aspect of the proof. Han’s work gives students a chance to check their understanding of the situation and what is being asked of them, as Han is struggling to articulate what is given and what relationships can be seen in the diagram. Clare’s work reminds students that the given triangle is isosceles and mentions the angles that we want to prove are congruent. Andre’s work mentions congruent triangles that can be used to establish that the desired angles are corresponding parts of congruent triangles.
Arrange students in groups of 2.
Design Principle(s): Support sense-making
Han, Clare, and Andre thought of a way to construct an angle bisector. They used a circle to construct points \(D\) and \(E\) the same distance from \(A\). Then they connected \(D\) and \(E\) and found the midpoint of segment \(DE\). They thought that ray \(AF\) would be the bisector of angle \(DAE\). Mark the given information on the diagram:
Han’s rough-draft justification: \(F\) is the midpoint of segment \(DE\). I noticed that \(F\) is also on the perpendicular bisector of angle \(DAE\).
Clare’s rough-draft justification: Since segment \(DA\) is congruent to segment \(EA\), triangle \(DEA\) is isosceles. \(DF\) has to be congruent to \(EF\) because they are the same length. So, \(AF\) has to be the angle bisector.
Andre’s rough-draft justification: What if you draw a segment from \(F\) to \(A\)? Segments \(DF\) and \(EF\) are congruent. Also, angle \(DAF\) is congruent to angle \(EAF\). 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 \(AF\) must be an angle bisector.
If students struggle to come up with reasons for their proof, suggest they look for structure by drawing auxiliary lines, marking the diagram, and using their reference chart.
Before students share their own proof, invite students to share good ideas they heard in Clare and Andre’s rough drafts. (Clare says the triangle is isosceles. Andre says there are congruent triangles.) Then ask what information they need to write a complete proof. (Three pairs of corresponding parts to write a triangle congruence proof.) Next, invite students to self-assess whether their proof has enough statements and reasons. Finally, invite a student who thinks they have a complete proof to share with the class.
Supports accessibility for: Language; Social-emotional skills; Attention
14.3: Reflecting on Reflection (15 minutes)
This optional activity focuses on critiquing the reasoning of others (MP3). Students read and summarize a valid proof that an isosceles triangle has symmetry using the angle bisector of the vertex angle as a line of reflection. Then they critique a false proof that a parallelogram has symmetry using the diagonal as a line of reflection. This gives students an opportunity to experience a more complex proof, and also to understand how the given statements in a proof influence what can be concluded.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Supports accessibility for: Language; Conceptual processing
Here is a diagram of an isosceles triangle \(APB\) with segment \(AP\) congruent to segment \(BP\).
Here is a valid proof that the angle bisector of the vertex angle of an isosceles triangle is a line of symmetry.
- 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 \(AP\) is congruent to segment \(BP\) because triangle \(APB\) is isosceles.
- The angle bisector of \(APB\) intersects segment \(AB\). Call that point \(Q\).
- By the definition of angle bisector, angles \(APQ\) and \(BPQ\) are congruent.
- Segment \(PQ\) is congruent to itself.
- By the Side-Angle-Side Triangle Congruence Theorem, triangle \(APQ\) must be congruent to triangle \(BPQ\).
- Therefore the corresponding segments \(AQ\) and \(BQ\) are congruent and corresponding angles \(AQP\) and \(BQP\) are congruent.
- Since angles \(AQP\) and \(BQP\) are both congruent and supplementary angles, each angle must be a right angle.
- So \(PQ\) must be the perpendicular bisector of segment \(AB\).
- Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the triangle across \(PQ\) the vertex \(P\) will stay in the same spot and the 2 endpoints of the base, \(A\) and \(B\), will switch places.
- Therefore the angle bisector \(PQ\) is a line of symmetry for triangle \(APB\).
Here is a diagram of parallelogram \(ABCD\).
Here is an invalid proof that a diagonal of a parallelogram is a line of symmetry.
- Read the proof and annotate the diagram with each piece of information in the proof.
- 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 \(M\).
- The diagonals of a parallelogram bisect each other, so \(MB\) is congruent to \(MD\).
- By the definition of parallelogram, the opposite sides \(AB\) and \(CD\) are parallel.
- Angles \(ABM\) and \(ADM\) are alternate interior angles of parallel lines so they must be congruent.
- Segment \(AM\) is congruent to itself.
- By the Side-Angle-Side Triangle Congruence Theorem, triangle \(ABM\) is congruent to triangle \(ADM\).
- Therefore the corresponding angles \(AMB\) and \(AMD\) are congruent.
- Since angles \(AMB\) and \(AMD\) are both congruent and supplementary angles, each angle must be a right angle.
- So \(AC\) must be the perpendicular bisector of segment \(BD\).
- Because reflection across perpendicular bisectors takes segments onto themselves and swaps the endpoints, when we reflect the parallelogram across \(AC\) the vertices \(A\) and \(C\) will stay in the same spot and the 2 endpoints of the other diagonal, \(B\) and \(D\), will switch places.
- Therefore diagonal \(AC\) is a line of symmetry for parallelogram \(ABCD\).
Are you ready for more?
There are quadrilaterals for which the diagonals are lines of symmetry.
- What is an example of such a quadrilateral?
- How would you modify this proof to be a valid proof for that type of quadrilateral?
Select students to share their annotations on the parallelogram diagram. Invite students to share the errors that they found and explain why they are errors.
Navigate to this URL: ggbm.at/xR2pAwz9. Display the applet in its starting position.
Here are some questions for discussion:
- “If I told you segment \(FA\) was a line of symmetry of this triangle, what else could you tell me about that line?” (It’s the perpendicular bisector of \(DE\). It’s the angle bisector of angle \(DAE\). \(F\) is the midpoint of \(DE\). If you reflect across it the triangle stays in the same place and \(D\) and \(E\) swap positions).
- “Can you justify your answers using transformations or congruent parts of congruent figures?” (Possible justification of why \(FA\) is an angle bisector: Since \(FA\) is a line of symmetry, when you reflect across \(FA\) everything coincides, so angles \(EAF\) and \(DAF\) must be congruent.)
- “If triangle \(ADE\) were not isosceles, would it still have a line of symmetry?” (No, because when you reflected the two sides wouldn’t coincide perfectly if they were different lengths.)
- “If triangle \(ADE\) were not isosceles, would the angle bisector of \(DAE\) and the perpendicular bisector of \(DE\) be the same line?” (Not necessarily.)
Using the applet, move \(D\) or \(E\) so that the triangle is no longer isosceles. Invite students to tell you:
- which line is the angle bisector of angle \(DAE\)
- which line goes from \(A\) to the midpoint of \(DE\)
- which line is perpendicular to \(DE\) and goes through \(A\).
Show students the image from Han, Clare, and Andre’s angle bisector construction and ask students to reflect on why they used a circle. (To make sure \(D\) and \(E\) were the same distance from \(A\).)
14.4: Cool-down - Going Both Ways Again (10 minutes)
Student Lesson Summary
Earlier we constructed an angle bisector, but we did not prove that the construction always works. Now that we know more we can see why each step is necessary for the construction to precisely bisect an angle. The proof uses some ideas from constructions:
- The midpoint of a segment divides the segment into 2 congruent segments.
- All the radii of a given circle are congruent.
But it also uses some ideas from triangle congruence:
- If triangles have 2 pairs of sides and the angle between them congruent, then the triangles are congruent.
- If triangles are congruent, then the corresponding parts of those triangles are also congruent.
Triangle congruence theorems and properties of rigid transformations can be useful for proving many things, including constructions.