Lesson 11What Is the Same?

Learning Goal

Let’s decide whether shapes are the same.

Learning Targets

  • I can decide visually whether or not two figures are congruent.

Lesson Terms

  • congruent

Warm Up: Find the Right Hands

Problem 1

A person’s hands are mirror images of each other. In the diagram, a left hand is labeled. Shade all of the right hands.

7 Drawings of hands - some left hands and some right hands. A left hand is labeled.

Activity 1: Are They the Same?

Problem 1

For each pair of shapes, decide whether or not they are the same.

  1. A drawing of a lightning bolt type figure and an upside down version of it.
  2. Two parallelograms with different dimensions.
  3. Two sets of partial spirals
  4. A polygon and a similar shaped but different sized polygon
  5. Two puzzle pieces with the notch in different places

Activity 2: Area, Perimeter, and Congruence

Problem 1

Rectangles of different sizes with dimensions labeled A-F

  1. Which of these rectangles have the same area as Rectangle but different perimeter?

  2. Which rectangles have the same perimeter as Rectangle but different area?

  3. Which have the same area and the same perimeter as Rectangle ?

  4. Use materials from the geometry tool kit to decide which rectangles are congruent. Shade congruent rectangles with the same color.

Are you ready for more?

Problem 1

In square , points , , , and are midpoints of their respective sides. What fraction of square is shaded? Explain your reasoning.

A square labeled ABCD. Four lines crisscross the interior to midpoints in the square making a blue square in the middle.

Lesson Summary

Congruent is a new term for an idea we have already been using. We say that two figures are congruent if one can be lined up exactly with the other by a sequence of rigid transformations. For example, triangle is congruent to triangle because they can be matched up by reflecting triangle across followed by the translation shown by the arrow. Notice that all corresponding angles and side lengths are equal.

Triangle ABC with side lengths and angles labeled. Similar triangle DEF can be moved from point F to point B' (reflected from ABC) to match up.

Here are some other facts about congruent figures:

  • We don’t need to check all the measurements to prove two figures are congruent; we just have to find a sequence of rigid transformations that match up the figures.

  • A figure that looks like a mirror image of another figure can be congruent to it. This means there must be a reflection in the sequence of transformations that matches up the figures.

  • Since two congruent polygons have the same area and the same perimeter, one way to show that two polygons are not congruent is to show that they have a different perimeter or area.