# ____
NAME _________________________________
Gr8 – MOD 2 – Lesson 11
8.G.A.2 Understand that a 2-dimensional figure is congruent to another if the second can be obtained from the first
by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence
that exhibits the congruence between them.
STUDENT OUTCOMES
 Students know the definition of congruence and related notation. Students know that to prove two figures are
congruent there must be a sequence of rigid motions that maps one figure onto the other.
 Students know the basic properties of congruence are similar to the properties for all three rigid motions
(translations, rotations, and reflections).
NOTES

A geometric figure 𝑆 is said to be congruent to another geometric figure 𝑆 ′
if there is a sequence of rigid motions that maps 𝑆 to 𝑆 ′ .

The notation related to congruence is the symbol ≅.
When two figures are congruent, like 𝑆 and 𝑆′, we can write: 𝑆 ≅ 𝑆 ′ .
Figure S
Figure S’
EXERCISE 1
Describe the sequence of rigid motions that demonstrates the two triangles shown below are congruent.
Note: A sequence to demonstrate congruence can be made up of any combination of the basic rigid
motions using all three or even just one.
1
EXERCISE 2
Describe a sequence of basic rigid motions (a congruence) that shows 𝑆1 ≅ 𝑆2 .
S2
S3
S1
Describe the sequence of basic rigid motions that shows 𝑆2 ≅ 𝑆3 .
Describe a sequence of basic rigid motions that shows 𝑆1 ≅ 𝑆3 .
Are all three figures ≅ congruent?
If figure 𝑆1 ≅ 𝑆3 and 𝑆2 ≅ 𝑆3 , what does this prove?
All 3 figures are _________________ since there is a sequence of b_________ r________ m__________
that ____________ each figure to each of the other figures.
2
EXERCISE 3
Perform the sequence of a translation followed by a reflection of Figure 𝑋𝑌𝑍, where 𝑇 is a translation along a
vector ⃗⃗⃗⃗⃗
𝐴𝐵 , and r is a reflection across line XY. Label the transformed figure 𝑋 ′ 𝑌 ′ 𝑍 ′ .
Will 𝑋𝑌𝑍 ≅ 𝑋 ′ 𝑌 ′ 𝑍 ′ ?
How do you know?
Y
B
X
Z
A
How long is side XZ?
How long is side X’Z’?
What is the measure of angle YXZ?
What is the measure of angle Y’X’Z’?
3
PRACTICE PROBLEMS
1) Are the two right triangles shown below congruent?
If so, describe a congruence (a sequence – one or more basic rigid motions) that would map one triangle
onto the other.
R
A
S
T
2) Given two rays, ⃗⃗⃗⃗⃗
𝑂𝐴 and ⃗⃗⃗⃗⃗⃗⃗⃗
𝑂′𝐴′:
A
O
A’
O’
Describe a congruence that maps ⃗⃗⃗⃗⃗
𝑂𝐴 to ⃗⃗⃗⃗⃗⃗⃗⃗
𝑂′𝐴′.
Describe a congruence that maps ⃗⃗⃗⃗⃗⃗⃗⃗
𝑂′𝐴′ to ⃗⃗⃗⃗⃗
𝑂𝐴.
4
3) Is △ 𝐴𝐵𝐶 ≅ △ 𝐴′ 𝐵 ′ 𝐶 ′ ?
If so, describe a sequence of rigid motions that proves they are congruent.
5 cm
If not, explain how you know.
5 cm
10 cm
13 cm
13 cm
10 cm
4) Is △ 𝐴𝐵𝐶 ≅ △ 𝐴′ 𝐵 ′ 𝐶 ′ ?
If so, describe a sequence of rigid motions that proves they are congruent.
If not, explain how you know.
5 cm
12 cm
5 cm
5 cm
Lesson Summary
Given that sequences have the same basic properties that basic rigid motions have, we can state these three basic
properties of congruences:
 A congruence maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
 A congruence preserves lengths of segments.
 A congruence preserves measures of angles.
The notation used for congruence is ≅.
5