# Lesson 9

Side-Side-Side Triangle Congruence

### Lesson Narrative

In this lesson, students complete their proofs of the triangle congruence theorems, studying the Side-Side-Side Triangle Congruence Theorem. They then have the opportunity to apply the theorem to a proof about parallelograms. Students continue to work on writing clear proofs. Students convince a partner and then a skeptic of their proof that opposite angles in a parallelogram are congruent (MP3). The second skill that is introduced in this lesson is looking for structure in order to decide which triangle congruence theorem applies in a particular proof (MP7). In the cool-down, students have an opportunity to study diagrams and try to identify which triangle congruence theorem applies. Each diagram they study is drawn from a proof of an important result, preparing students for later work proving theorems about triangles, parallelograms, and lines.

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.

### Learning Goals

Teacher Facing

• Justify (in writing) the Side-Side-Side Triangle Congruence Theorem.
• Prove (in writing) opposite angles of a parallelogram are congruent.

### Student Facing

• Let’s see if we can prove one more set of conditions that guarantee triangles are congruent, and apply theorems.

### Required Preparation

Provide students with dried linguine pasta if they request it, or tuck it unobtrusively into their geometry toolkits before class.

### Student Facing

• I can explain why the Side-Side-Side Triangle Congruence Theorem works.
• I can use the Side-Side-Side Triangle Congruence Theorem in a proof.

Building On

Building Towards

### Glossary Entries

• auxiliary line

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.

• converse

The converse of an if-then statement is the statement that interchanges the hypothesis and the conclusion. For example, the converse of "if it's Tuesday, then this must be Belgium" is "if this is Belgium, then it must be Tuesday."

• corresponding

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 $$EFD$$.

• parallelogram

A quadrilateral in which pairs of opposite sides are parallel.