**Problem 1 :**

Pentagons JKLMN and STUVW shown below are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

**Problem 2 : **

Decide whether the figures are similar. If they are similar, write a similarity statement.

**Problem 3 :**

We are asked to create a poster to advertise a field trip to see the Liberty Bell. We have a 3.5 inch by 5 inch photo that you want to enlarge. We want the enlargement to be 16 inches wide. How long will it be ?

**Problem 4 :**

The rectangular patio around a pool is similar to the pool as shown below. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.

**Problem 5 : **

In the diagram shown below, quadrilateral JKLM is similar to quadrilateral PQRS. Find the value of z.

**Problem 1 :**

Pentagons JKLMN and STUVW shown below are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

**Solution : **

Because JKLM ∼ STUVW, we can write

∠J ≅ ∠S

∠K ≅ ∠T

∠L ≅ ∠U

∠M ≅ ∠V

∠N ≅ ∠W

We can write the statement of proportionality as follows :

JK / ST = KL / TU = LM / UV = MN / VW = NJ / WS

**Problem 2 : **

Decide whether the figures are similar. If they are similar, write a similarity statement.

As shown, the corresponding angles of ABCD and EFGH are congruent. Also, the corresponding side lengths are proportional.

AB / EF = 15 / 10 = 3 / 2

BC / FG = 6 / 4 = 3 /2

CD / GH = 9 /6 = 3 / 2

DA / HE = 12 / 8 = 3 / 2

Hence, the two figures are similar and we can write

ABCD ∼ EFGH

**Note : **

If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. In the above example, the common ratio is 3/2 is the scale factor of ABCD to EFGH.

**Problem 3 :**

We are asked to create a poster to advertise a field trip to see the Liberty Bell. We have a 3.5 inch by 5 inch photo that you want to enlarge. We want the enlargement to be 16 inches wide. How long will it be ?

**Solution : **

To find the length of the enlargement, you can compare the enlargement to the original measurements of the photo.

From the diagram shown above, we have

x in. / 5 in. = 16 in. / 3.5 in.

5 ⋅ (x in. / 5 in. = 16 in. / 3.5 in.) ⋅ 5

Simplify.

x ≈ 22.9 in.

Hence, the length of the enlargement will be about 23 inches.

**Problem 4 :**

The rectangular patio around a pool is similar to the pool as shown below. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.

**Solution : **

Because the rectangles are similar, the scale factor of the patio to the pool is 48 ft : 32 ft, which is 3 : 2 in simplified form.

The perimeter of the patio is

2(24) + 2(48) = 144 feet

and the perimeter of the pool is

2(16) + 2(32) = 96 feet.

The ratio of the perimeters is

= 144 / 96

= 3 / 2

In similar figures, the ratio of the perimeters is the same as the scale factor.

So, the scale factor of the patio to the pool is 3/2.

**Problem 5 : **

In the diagram shown below, quadrilateral JKLM is similar to quadrilateral PQRS. Find the value of z.

**Solution : **

Because the quadrilaterals JKLM is similar to PQRS, we can set up the proportion that contains PQ.

Write proportion :

KL / QR = JK / PQ

Substitute.

15 / 6 = 10 / z

5 / 2 = 10 / z

Using reciprocal property,

2 / 5 = z / 10

or

z / 10 = 2 / 5

Multiply each side by 10.

10 ⋅ (z / 10) = 10 ⋅ (2 / 5)

Simplify.

z = 4

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