Section 5.6 Overlapping Triangles

We'll begin with a review of Side-Side-Side, Side-Angle-Side, and Angle-Side-Angle Congruence Postulates!

In Section 5.5, we talked about how to prove triangles are congruent three different ways.

One of them was by Side Side Side Congruence.

Remember, if three sides in one triangle are congruent to three sides in second triangle, then the triangles are congruent.

We will do an example using SSS Congruence where two triangles have corresponding same side lengths.

Remember that ORDER MATTERS when labeling triangles congruent. Make sure you match up corresponding sides.

Another was the Side-Angle-Side Congruence Postulate that stated that if two sides and an included angle in one triangle are congruent to two sides and an included angle in a second triangle, then the triangles are congruent.

We will do a couple of examples on the board with one involving the use of vertical angles as the included angle. Students will have to pick out these angles being congruent without them being given.

The third way we have learned so far to prove triangles congruent was by Angle-Side-Angle Congruence. If two angles and an included side in one triangle are congruent to two angles and an included side of a second triangle, then the triangles are congruent.

We will do some examples for each one of the above cases on the board or overhead.

We will practice labeling the triangles congruent in the proper order.


So far, we have been looking at congruent triangles that are sitting next to each other. Now, we're going to learn to recognize and use overlapping congruent triangles. This is what Section 5.6 covers in your book.

What does it mean for something to overlap?

Turn with me to page 282 and let's do the activity together on this page. Can someone read the introduction to the problem for me?

Problem: Kyle and his father own a print shop. Often customers come in with the request to have their company logo created and reproduced for their stationery. The customer depends on Kyle and his father to make a technically correct rendering of it.

Suppose a customer wants a logo to look like the start pattern above. In what ways can Kyle use a simpler geometric shape to create the logo?

Materials for Activity:

Each student will be given three different triangles, a large sheet of construction paper, and a pentagon.

Using the logo pattern in the book, students will reproduce the pattern using three different triangles. Each triangle will be traced and overlapped to form the logo pattern. The students will see how the use of overlapping congruent triangles can reproduce the star logo pattern that Kyle needs for his customer. The pentagon's purpose is to help the students visualize how to make the triangles work. It will be helpful if students will first trace the pentagon and then begin working with the triangles. However, the pentagon isn't required to make the construction.

After completion of the activity:

Does anyone know what a line of symmetry is?

Visualize having a figure drawn on a sheet of paper. Is it possible to fold the paper in half so that what part of the figure on each side is equivalent?

We will practice finding the line of symmetry in several figures that involve overlapping congruent triangles. This will help students be able to pick out the congruent triangles and corresponding sides and angles. It will also help them with labeling the triangles congruent in the correct order.

I will summarize what we learned in class today about how overlapping congruent triangles can be used to form figures and designs and how when labeling triangles congruent, the ORDER IS VERY IMPORTANT.


Students will be asked to work on a Section 5.6 worksheet for homework to give them practice picking out congruent triangles when they overlap and writing the order correctly.