Topics
5. Congruence and Triangles
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United States of America
Grades 9-12

5.01 Congruence transformations

Lesson
Practice

Introduction

In 8th grade, we saw the congruency in figures and formalized that in this course with translations, reflections, and rotations. We will continue to use transformations to justify congruency between figures in this lesson.

Using transformations to justify congruence

Exploration

Consider triangle ABC with coordinates A \left(-7,6 \right), B \left(5,6 \right), and C \left(-1,12 \right). Consider the following transformations:

  • A reflection across the line y=x

  • A translation to the right 4 units and down 7 units

  • A rotation 270 \degree clockwise about the origin

  1. What prediction can you make about how the segment lengths of the triangle changes when performing the transformations alone or in sequence?

  2. What prediction can you make about how the angle measures of the triangle changes when performing the transformations alone or in sequence?

Congruent

Two figures are congruent if and only if there is a rigid transformation or sequence of transformations that maps one of the figures onto the other. Congruent figures will have the same shape and size.

A four quadrant coordinate plane with two triangles plotted at Quadrants 1 and 4. Triangle A B C is plotted with Point A at ( negative 3, 4), point B at ( negative 3, 1) and point C at ( negative 1, 1). Triangle D E F is plotted at quadrant 4 with point D at (1, negative 1), E at (3, negative 1) and F at (1, negative 4).

\triangle{ABC}\cong\triangle{FDE} because a translation 4 units to the right followed by a reflection across the x-axis will map \triangle ABC to \triangle DEF.

Examples

Example 1

The triangles in the diagram are congruent.

Triangle A B C and triangle P Q R are drawn such that segment A B and segment P R are marked congruent, as well as segment B C and segment Q R. Segment A C and segment P Q are also marked congruent.
a

Identify the transformations that map one triangle to its image.

Worked Solution
Create a strategy

There are three rigid transformations that preserve length and angle measure: translations, reflections, and rotations. We will draw a diagram to determine which transformations can be applied to map one triangle onto the other.

Triangle A B C reflected across a line resulting to triangle A prime B prime C prime. Triangle A prime B prime C prime is translated to the right resulting to triangle P Q R. A B, A prime B prime, and P R are congruent. B C, B prime C prime, and Q R are congruent. A C, A prime C prime, and P Q are congruent.
Apply the idea

A reflection and a translation will map the first figure onto the other.

Reflect and check

It looks like a rotation would map the triangles onto each other, but after rotating one triangle the congruency marks do not correspond correctly.

b

Complete the congruency statement: \triangle{ABC}\cong \triangle{⬚}.

Worked Solution
Create a strategy

Identify the vertices that correspond to each A, B, and C, then put them in the same order.

Apply the idea

Since A mapped to P, B mapped to R, and C mapped to Q, we can say \triangle ABC \cong \triangle PRQ.

Reflect and check

The order of a congruency statement matters. The vertices in the pre-image need to be in the same order as the vertices to which they map in the image.

Example 2

Consider the two figures on the coordinate grid shown:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y

Determine whether the figures are congruent. If so, justify their congruency using a sequence of transformations.

Worked Solution
Create a strategy

Since the orientation of the image has changed, we should consider corresponding sides to decide if a reflection or a rotation has occurred.

The shortest sides \overline{BC} and \overline{QV} seem to be corresponding and are turned, so a rotation may have occurred. By testing one of the points, we can determine if a rotation is the only transformation. If A was rotated 90 \degree clockwise, the image would be located at the point (1,4), which is 2 units above the actual image S.

A rotation and a translation could map the pre-image to the image. Confirm the transformations using the coordinates and determine if the figures are congruent.

Apply the idea

We can attempt to map \triangle ABC to \triangle SQV by applying a rotation of 90 \degree clockwise about the origin and then a translation 2 units down.

First, a rotation 90 \degree about the origin, (x, y) \to (y, -x), would lead to the figure:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
  • A(-4,1) \to (1, 4)
  • B(-2,4) \to (4, 2)
  • C(0,3) \to (3, 0)

Then, a translation 2 units down (x, y) \to (x, y-2), would lead to the figure:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
  • (1, 4) \to S(1, 2)
  • (4, 2) \to Q(4,0)
  • (3, 0) \to V(3, -2)

Since the sequence of transformations that maps \triangle ABC onto \triangle SQV is a rotation 90 \degree about the origin followed by a translation 2 units down, we can state that the figures are congruent because rotations and translations are rigid transformations.

Reflect and check

We can combine the coordinate mappings of the sequence into one mapping: (x,y) \to (y, -x-2)

Example 3

Identify any congruent figures on the coordinate plane. Justify your reasoning.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Worked Solution
Create a strategy

By determining if there is a rigid transformation or sequence of rigid transformations that will map one figure onto another, we can confirm congruency between figures.

Apply the idea

Rectangle KLMN is a translation of rectangle FEGD \ 4 units up and 3 units right. So, rectangle KLMN \cong rectangle FEGD.

\triangle POQ is a rotation of \triangle BAC \ 90 \degree counterclockwise about the origin. So, \triangle POQ \cong \triangle BAC.

\triangle SRT is a reflection of \triangle IHJ across the y-axis. So, \triangle SRT \cong \triangle IHJ.

Idea summary

Two figures are congruent if and only if there is a rigid transformation or sequence of transformations that maps one of the figures onto the other.

We can show congruency by identifying a rigid transformation or a sequence of rigid transformations that map one figure onto the other.

Outcomes

G.CO.A.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

G.CO.B.6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

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