Lesson 13Congruence

Learning Goal

Let’s find ways to test congruence of interesting figures.

Learning Targets

I can use distances between points to decide if two figures are congruent.

Lesson Terms

congruent
right angle

Warm Up: Not Just the Vertices

Trapezoids $A B C D$ and $A^{'} B^{'} C^{'} D^{'}$ are congruent.

Draw and label the points on $A^{'} B^{'} C^{'} D^{'}$ that correspond to $E$ and $F$ .
Draw and label the points on $A B C D$ that correspond to $G^{'}$ and $H^{'}$ .
Draw and label at least three more pairs of corresponding points.

Trapezoid ABCD with point E between A and D and point F between C and D. Trapezoid A'B'C'D' with point G' between A' and D' and point H' between A' and B'

Activity 1: Congruent Ovals

Are any of the ovals congruent to one another? Explain how you know.

Are you ready for more?

You can use 12 toothpicks to create a polygon with an area of five square toothpicks, like this:

A plus sign on a grid with 1 unit sides.

Can you use exactly 12 toothpicks to create a polygon with an area of four square toothpicks?

Activity 2: Corresponding Points in Congruent Figures

Here are two congruent shapes with some corresponding points labeled.

Draw the points corresponding to $B$ , $D$ , and $E$ , and label them $B^{'}$ , $D^{'}$ , and $E^{'}$ .
Draw line segments $A D$ and $A^{'} D^{'}$ and measure them. Do the same for segments $B C$ and $B^{'} C^{'}$ and for segments $A E$ and $A^{'} E^{'}$ . What do you notice?
Do you think there could be a pair of corresponding segments with different lengths? Explain.

Activity 3: Astonished Faces

Are these faces congruent? Explain your reasoning.

Lesson Summary

To show two figures are congruent, you align one with the other by a sequence of rigid transformations. This is true even for figures with curved sides. Distances between corresponding points on congruent figures are always equal, even for curved shapes. For example, corresponding segments $A B$ and $A^{'} B^{'}$ on these congruent ovals have the same length:

Two congruent ovals on a square grid. In the first oval, two points on opposite sides of the oval are labeled A and B and are connected by a line segment. In the second oval, two points on opposite sides of the oval are labeled A prime and B prime and are connected by a line segment.

To show two figures are not congruent, you can find parts of the figures that should correspond but that have different measurements.

For example, these two ovals don’t look congruent.

On both, the longest distance is 5 units across, and the longest distance from top to bottom is 4 units. The line segment from the highest to lowest point is in the middle of the left oval, but in the right oval, it’s 2 units from the right end and 3 units from the left end. This proves they are not congruent.

The same two ovals with perpendicular lines drawn in

Lesson 13Congruence

Learning Goal

Learning Targets

Lesson Terms

Warm Up: Not Just the Vertices

Problem 1

Activity 1: Congruent Ovals

Problem 1

Are you ready for more?

Problem 1

Activity 2: Corresponding Points in Congruent Figures

Problem 1

Activity 3: Astonished Faces

Problem 1

Lesson Summary