NAME _____________________________________________ DATE ____________________________ PERIOD _____________
LT 7.1 Reteach
Congruence and Transformations
Translation
Reflection
Rotation
• length is the same
• length is the same
• length is the same
• orientation is the same
• orientation is different
• orientation is different
Example
Determine if the two figures are congruent by using transformations.
The two triangles are congruent because a
rotation followed by a translation will map
∆XYZ onto ∆RST.
Exercises
Determine if the two figures are congruent by using transformations.
1.
Course 3 • Chapter 7 Congruence and Similarity
2.
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
LT 7.2 Reteach
Congruence
If two figures are congruent, their corresponding sides are congruent and their corresponding angles are congruent.
Example
Write congruence statements comparing the corresponding parts in the congruent triangles shown.
Use the matching arcs and tick marks to identify the corresponding parts.
Corresponding angles:
∠ S ≅ ∠ R, ∠ A ≅ ∠ B, ∠ T ≅ ∠ X
Corresponding sides:
̅̅̅̅
̅̅̅ ≅ ̅̅̅̅
𝑆𝐴 ≅ ̅̅̅̅
𝑅𝐵, ̅̅̅̅̅
𝐴𝑇 ≅ ̅̅̅̅
𝐵𝑋, ̅𝑇𝑆
𝑋𝑅
Exercises
Write congruence statements comparing the corresponding parts in the congruent figures shown.
1.
2.
3.
4.
Course 3 • Chapter 7 Congruence and Similarity
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
LT 7.3 Reteach
Similarity and Transformations
Two figures are similar if the second can be obtained from the first by a sequence of transformations and dilations.
Recall that a dilation changes the size of a figure by a scale factor, but does not change the shape of the figure.
Example 1
Determine if the two figures are similar by using
transformations.
Since the orientation of the figures is the same, one of the
transformations is a translation.
Write ratios comparing the lengths of the sides.
𝐴𝐵
𝑅𝑆
=
4
8
or
1
2
𝐵𝐶
, 𝑆𝑇 =
2
4
1 𝐶𝐷
or 2 , 𝑇𝑈 =
4
8
1
or 2 , =
𝐷𝐴
𝑈𝑅
=
2
4
or
1
2
Since the ratios are equal, ABCD is the dilated image of RSTU.
So, the two triangles are similar because a translation and a
dilation maps ABCD onto RSTU.
Example 2
Determine if the two figures are similar by using
transformations.
Since the orientation of the figures is the same, one of the
transformations.
𝐴𝐵
𝑅𝑆
=
3 𝐵𝐶
7 , 𝑆𝑇
=
2
4
or
1
2
The ratios are not equal. So, the two triangles are not similar
since a dilation did not occur.
Exercise
1. Determine if the two figures are similar by using
transformations.
Course 3 • Chapter 7 Congruence and Similarity
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
LT 7.4 Reteach
Properties of Similar Polygons
Two polygons are similar if they have the same shape. If the polygons are similar, then their corresponding angles
are congruent and the measures of their corresponding sides are proportional. Use the symbol ∼ for similarity.
Example 1
Determine whether ∆ABC is similar to ∆DEF. Explain.
∠ A ≅∠ D, ∠ B ≅ ∠ E, ∠ C ≅ ∠ F,
𝐴𝐵
𝐷𝐸
=
4
6
𝑜𝑟
2
3
,
𝐵𝐶
𝐸𝐹
=
6
9
𝑜𝑟
2
3
,
𝐴𝐶
𝐷𝐹
=
8
12
𝑜𝑟
2
3
The corresponding angles are congruent, and the
corresponding sides are proportional.
So, ∆ABC is similar to ∆DEF, or ∆ABC ~ ∆DEF.
Example 2
Given that polygon KLMN ~ polygon PQRS, find the missing measure.
Find the scale factor from polygon KLMN to polygon PQRS.
scale factor:
𝑃𝑆
𝐾𝑁
=
3
4
The scale factor is the
constant of proportionality.
3
A length on polygon PQRS is 4 times as long as a corresponding length
on polygon KLMN.
3
x = 4 (5)
x=
15
or
4
Write the equation.
3.75
Multiply.
Exercises
1. Determine whether the polygons
below are similar. Explain.
Course 3 • Chapter 7 Congruence and Similarity
2. The triangles below are similar.
Find the missing measure.
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
LT 7.5 Reteach
Slope and Similar Triangles
Recall that the slope of a line is the ratio of the rise to the run. You can use properties of similar triangles to show the
ratios of the rise to the run for each right triangle are equal.
Example
Write a proportion comparing the rise to the run for
each of the similar slope triangles shown at the right.
Then find the numeric value.
𝐶𝐵
𝐸𝐷
=
𝐵𝐴
𝐷𝐶
Corresponding sides of similar
triangles are proportional.
CB • DC = ED • BA
Find the cross products.
𝐶𝐵 •𝐷𝐶
𝐵𝐴 • 𝐷𝐶
= 𝐵𝐴 • 𝐷𝐶
Division Property of Equality
𝐶𝐵
𝐵𝐴
Simplify.
𝐸𝐷 • 𝐵𝐴
1
2
𝐶𝐵
=
𝐸𝐷
𝐷𝐶
3
=6
𝐸𝐷
CB = 1, BA = 2, ED = 3, DC = 6
1
3
So, 𝐵𝐴 = 𝐷𝐶 , or 2 = 6.
Exercises
1. Graph ∆XYZ with vertices X(–3, 5),
Y(–3, 3), and Z(0, 3) and ∆ZLP with
vertices Z(0, 3), L(0, –1), and P(6, –1).
Then write a proportion comparing the
rise to the run for each of the similar
slope triangles and find the numeric
value.
Course 3 • Chapter 7 Congruence and Similarity
2. Graph ∆ABE with vertices A(-4, -3),
B(0, 0), and E(0, -3) and ∆ACD with
vertices A(–4, –3), C(4, 3), and D(4, –3).
Then write a proportion comparing the
rise to the run for each of the similar
slope triangles and find the numeric
value.
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