Similarity and CongruenceWarm-Up

1) Solve for x. 2) Solve for x: x2−3 x−21=0

Congruent Figures

Two figures are congruent if they have the same shape and size. The tessellation at right is produced by replicating the same figure over and over. How do show mathematically that two figures are congruent?

Definition of Congruent Polygons

Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

The symbol for “is congruent to” is ≅

Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.

Model Problem Given polygon ABCD is congruent to polygon EFGH

The corresponding sides are congruent: The corresponding angles are congruent:

____________ _____________ _____________ ____________

____________ _____________ _____________ ____________

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Exercise Given triangle ABC is congruent to triangle DEF

List three pairs of congruent angles: List three pairs of congruent sides:

__________________________ ___________________________

__________________________ ___________________________

__________________________ ___________________________

Using Corresponding Parts of Congruent Triangles

Model Problem

Given: ∆ABC @ ∆DBC

a) Find the value of x.

b) Find the measure of angle DBC.

Exercise Given: ∆ABC @ ∆DEF

a) Find the value of x.

b) Find the measure of angle E.

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Similar Polygons

When we order different-sized prints from a photography studio, we expect the original image to be enlarged or reduced without distortion. For example, if we divide the width by 4 to make a smaller print, we must also divide the length by 4 also.

A photo and its enlargement are an example of similar polygons. Two figures that are similar have the same shape but not necessarily the same size.

Definition of Similar Polygons

Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional. The side lengths are proportional (or in proportion) if all pairs of corresponding sides are in the same ratio.

For example, the photos above are in proportion because both the length and width are divided by 4. We can say the sides of the largest photo and the smallest photo are in a 4:1 ratio.

The symbol for “is similar to” is ~

More Examples

Similar Figures Congruent Figures Neither Similar Nor Congruent

Critical Thinking

1) What gives congruent and similar figures their identical shapes?

2) Look at the above pictures of similar and congruent triangles. What part of the triangle makes congruent triangles the same size but similar figures different sizes?

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Using Corresponding Parts of Similar Polygons

Just like with congruent polygons, we can match up corresponding sides and angles.Remember that corresponding angles are congruent and corresponding sides are in proportion.

Given: ∆QRT ~ ∆NPM

Corresponding angles are CONGRUENT:

__________________________________

__________________________________

__________________________________

Corresponding sides are IN PROPORTION. We write this as equal ratios (fractions):

The similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons.

The similarity ratio of ∆ABC to ∆DEF is 36 , or

12 .

The similarity ratio of ∆DEF to ∆ABC is 63 , or 2.

When writing the similarity statement, be sure to write the congruent sides and angles in corresponding order.

Correct Way Incorrect WayTriangle VUT ~ triangle LKJ Triangle VUT ~ triangle JLK

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Model Problem A

Determine whether the given polygons are similar. If so, write the similarity ratio and a similarity statement.

1) Determine if corresponding sides are in proportion:

2) Determine if corresponding angles are congruent:

Model Problem B

Determine whether the given polygons are similar. If so, write the similarity ratio and a similarity statement.

1) Determine if corresponding sides are in proportion:

2) Determine if corresponding angles are congruent:

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Exercise

Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio and a similarity statement.

Lesson Quiz

1) Two figures are congruent if and only if their corresponding sides are _________________ and their corresponding angles are _____________.

2) Two figures are similar if and only if their corresponding sides are _________________ and their corresponding angles are _____________.

3) Draw two right triangles that are congruent, two right triangles that are similar but not congruent, and two right triangles that are neither similar nor congruent. Label sides and angles with numerical values.

4) Determine if the following figures are similar. Show mathematically why or why not.

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Homework

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Similar Polygon Problems

Warm-up

1) Determine if the following polygons are similar.

2) Multiply and express your answer as a trinomial: (x – 2)(x + 6)

Solving Proportions

Recall that in similar figures, the corresponding sides are in proportion. In a proportion, the cross-products are equal. That is, when

ab= c

d we have ad = bc

Model Problems

A) A tree 24 feet tall casts a shadow 16 feet long at the same time a man 6 feet tall casts a shadow x feet long. What is the length of the man’s shadow?

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B) In the accompanying diagram, is similar to , , and , if AB = 3, BC = 12, DE = x + 2, and EF = 18, find the value of x.

C)

Exercises

1) Find the length of the model to the nearest tenth of a centimeter.

2) A triangle has sides of length 3, 5, and 7. In a similar triangle, the shortest side has a length of x – 3, and the longest side has a length of x + 5. Find the value of x.

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3) Two triangles are similar. If the lengths of the sides of the larger triangle are 12, 18, and 24 and the length of the shortest side of a similar triangle is 4, what is the perimeter of the smaller triangle?

More Similar Triangle Problems

Model Problem A: Using Quadratics

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Model Problem B: Twisted Triangles

Exercise

1)

2)

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Homework

6)

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