In previous lessons, students developed a sequence of rigid motions that would always work to take one congruent triangle onto another. Additionally, students proved that any pair of segments with the same length is congruent. In this lesson, students use these ideas to prove the Side-Angle-Side Triangle Congruence Theorem. That is, they justify that if you know that two pairs of corresponding sides and the pair of corresponding angles between those sides are congruent, then there must be a sequence of rigid motions that takes one triangle exactly onto the other. Students are then given the opportunity to apply the theorem to prove the base angles are congruent in isosceles triangles.
Applying the triangle congruence theorems in this and subsequent lessons often involves purposefully drawing additional lines to make triangles with certain properties. Drawing these auxiliary lines is an important way that mathematicians look for and make use of structure (MP7).
Technology isn‘t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
- Justify (in writing) the Side-Angle-Side Triangle Congruence Theorem.
- Prove (in writing) the Isosceles Triangle Theorem.
- Let’s use definitions and theorems to figure out what must be true about shapes, without having to measure all parts of the shapes.
In the cool-down, save students’ drawings. They will be useful for later lessons.
- I can explain why the Side-Angle-Side Triangle Congruence Theorem works.
- I can use the Side-Angle-Side 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 \(EFD\).