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

10.01 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:

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x
-5
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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:

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x
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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:

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-5
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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.

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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 4 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.

Example 4

Consider the figure shown on the graph below. It can be used as an example to show what happens to the segment lengths and angle measures after different types of transformations.

2
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a

Translate ABCD to the right 3 units, and then find the segment lengths and angle measures for both the pre-image and its image.

Worked Solution
Apply the idea

After we translate ABCD 3 units to the right we get this image:

2
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x
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y

We know that two of the side lengths of ABCD and A'B'C'D' are horizontal because the y-coordinates of A and D are both zero, and the y-coordinates of A' and D' are both zero. The y-coordinates of B and C are both 2, and the y-coordinates of B' and C' are both 2.

We know that two of the side lengths of ABCD and A'B'C'D' are vertical because the x-coordinates of A and B are both zero, and the x-coordinates of A' and B' are both 3. The x-coordinates of C and D are both 2, and the x-coordinates of C' and D' are both 5.

This means the segments are perpendicular and all angles on both figures are 90 \degree.

In order to find the segment lengths of the horizontal sides, \overline{AD} and \overline{BC}, we only need to subtract the x-coordinates. 2-0 =2 \text{ units}

In order to find the segment lengths of the horizontal sides, \overline{A'D'} and \overline{B'C'}, we only need to subtract the x-coordinates. 5-3 =2 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{BA} and \overline{CD}, we only need to subtract the y-coordinates. 2-0 =2 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{B'A'} and \overline{C'D'}, we only need to subtract the y-coordinates. 2-0 =2 \text{ units}

Reflect and check

A translation to the right 3 units has a transformation mapping: (x, y) \to (x+3, y)

b

Figure LMNP is the image of figure ABCD after a dilation. Find the segment lengths and angle measures for the image.

-4
-2
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-4
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Worked Solution
Apply the idea

We know that two of the side lengths of LMNP are horizontal because the y-coordinates of L and P are both -3. The y-coordinates of M and N are both 1.

We know that two of the side lengths of LMNP are vertical because the x-coordinates of L and M are both -4. The x-coordinates of N and P are both 0.

This means the segments are perpendicular and all angles are 90 \degree.

In order to find the segment lengths of the horizontal sides, \overline{LP} and \overline{MN}, we only need to subtract the x-coordinates. 0-(-4) =4 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{ML} and \overline{NP}, we only need to subtract the y-coordinates. 1-(-3) =4 \text{ units}

c

State the relationships between the corresponding sides and angles of the pre-image and image after the translation in part (a). Then state the relationships between the corresponding pre-image and image after the dilation in part (b).

Worked Solution
Create a strategy

Identify whether the transformations preserved distance, angles, or distance and angles.

Apply the idea

  • We can see that distance has been preserved when translating ABCD since the side lengths of the image are the same as the pre-image in part (a). The angle measurements are all 90 \degree. Therefore, a translation to the right 3 units preserves both distance and angle measurements.

  • We can see that distance has not been preserved when dilating ABCD since the side lengths of the image are twice as large as the pre-image in part (b). The angle measurements are all 90 \degree. Therefore, a dilation using a scale factor of 2 preserves only angle measurements.

Example 5

The following sequences of transformations are applied to figure ABCD. Without performing the transformations, determine whether each final image will be a similarity transformation, a congruency transformation, or both. Explain your reasoning.

a

A reflection across the x-axis followed by a rotation 90 \degree clockwise about the origin.

Worked Solution
Create a strategy

A reflection across the x-axis has a transformation mapping: (x,y) \to (x, -y)

A rotation 90 \degree clockwise about the origin has a transformation mapping: (x, y) \to (y, -x)

Apply the idea

Since a reflection across the x-axis has a transformation mapping (x,y) \to (x, -y), we know that the coordinates were not multiplied by a scale factor, meaning no dilation occurred.

After this rigid transformation, a rotation 90 \degree clockwise about the origin with a transformation mapping (x, y) \to (y, -x) also does not change the segments of the figure, since there is no scale factor other than 1.

The sequence of rigid transformations will change the orientation of ABCD, while preserving its angles and side lengths. Since the sequence of transformations will lead to a congruent image to the pre-image, the sequence is both a congruency transformation and a similarity transformation.

b

A dilation by a scale factor of \dfrac{1}{3} with the center of dilation at the origin followed by a rotation 180 \degree.

Worked Solution
Apply the idea

Since the scale factor k <1, any segment on the image will be k times the size of the line segments of the pre-image, so the side lengths will be proportional, but not congruent, while the angles will be preserved.

A rotation will preserve both the side lengths and angle measures of the figure, after it has been dilated.

The sequence of transformations will be a similarity transformation.

Reflect and check

The coordinate mapping for the sequence of transformations is (x,y) \to (-\dfrac{1}{3}x, -\dfrac{1}{3}y)

Multiplying the x- and y-coordinates by the same scale factor indicates a dilation, which creates a similar, but not congruent, transformation.

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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