Congruence and Transformations

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.

Course 3 • Chapter 7 Congruence and Similarity 99 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: , ,

Exercises Write congruence statements comparing the corresponding parts in the congruent figures shown.

Course 3 • Chapter 7 Congruence and Similarity 99 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.

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.

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 99 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,

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: The scale factor is the constant of proportionality. A length on polygon PQRS is times as long as a corresponding length on polygon KLMN. x = (5) Write the equation. x = or 3.75 Multiply.

Exercises

1. Determine whether the polygons 2. The triangles below are similar. below are similar. Explain. Find the missing measure.

Course 3 • Chapter 7 Congruence and Similarity 99 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.

= Corresponding sides of similar triangles are proportional.

CB • DC = ED • BA Find the cross products.

= Division Property of Equality

= Simplify.

= CB = 1, BA = 2, ED = 3, DC = 6

So, = , or = .

Exercises 1. Graph ∆XYZ with vertices X(–3, 5), 2. Graph ∆ABE with vertices A(-4, -3), Y(–3, 3), and Z(0, 3) and ∆ZLP with B(0, 0), and E(0, -3) and ∆ACD with vertices Z(0, 3), L(0, –1), and P(6, –1). vertices A(–4, –3), C(4, 3), and D(4, –3). Then write a proportion comparing the Then write a proportion comparing the rise to the run for each of the similar rise to the run for each of the similar slope triangles and find the numeric slope triangles and find the numeric value. value.

Course 3 • Chapter 7 Congruence and Similarity 99