In a previous lesson, students studied a proof of triangle congruence and used it as a model to write their own proof. In this lesson, they are asked to come up with most of the language for a proof of the Angle-Side-Angle Triangle Congruence Theorem (MP3). They are explicitly encouraged to use previous work and refer to the template as scaffolding.
Students are again given the opportunity to apply the theorem, proving opposite sides of parallelograms are congruent. Students use the definition: a parallelogram is a quadrilateral with two pairs of opposite sides parallel, alternate interior angles, and congruent triangles in their proof. If students struggled with the question on parallel lines and a transversal on the previous end of unit test, there is an optional activity to remind students that alternate interior angles formed by parallel lines and a transversal are congruent.
If some students use a transformation approach rather than applying the Angle-Side-Angle Triangle Congruence Theorem in the parallelogram proof, there is an opportunity to discuss how students can make use of repeated reasoning by using the Angle-Side-Angle Triangle Congruence Theorem rather than coming up with a different transformation proof for every case of two triangles that have two pairs of congruent corresponding angles and one pair of congruent corresponding sides between them (MP8).
One of the activities in this lesson works best when each student has access to internet-enabled devices because students will benefit from seeing the relationship in a dynamic way.
- Justify (in writing) the Angle-Side-Angle Triangle Congruence Theorem.
- Prove (in writing) opposite sides of a parallelogram are congruent.
- Let’s see if we can prove other sets of measurements that guarantee triangles are congruent, and apply those theorems.
Acquire internet-enabled devices that can run the applet for the activity Notice and Wonder: Assertion, one for every 2–3 students. If technology is not available, there is a paper and pencil alternative, but consider displaying the applet for all to see.
For the lesson synthesis: Create a display of tips for proving lines, angles, or triangles are congruent using triangle congruence theorems.
- I can explain why the Angle-Side-Angle Triangle Congruence Theorem works.
- I can use the Angle-Side-Angle Triangle Congruence Theorem in a proof.
An extra line drawn in a figure to reveal hidden structure.
For example, the line shown in the isosceles triangle is a line of symmetry, and the lines shown in the parallelogram suggest a way of rearranging it into a rectangle.
For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.
In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).
A quadrilateral in which pairs of opposite sides are parallel.