Section 7.3Similar Triangles

Example 1:

a) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Since mB = mD, B D.

By the Triangle Sum Theorem, 42° + 58° + mA = 180°, so mA = 80°.

Since mE = 80°, A E.

So, ΔABC ~ ΔEDF by the AA Similarity.

Example 1:

b) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

QXP NXM by the Vertical Angles Theorem.

Since QP || MN, Q N.

So, ΔQXP ~ ΔNXM by AA Similarity.

Triangle Similarity

Example 2: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

a)

9 3 6 3 7.5 3or , or , or

6 2 4 2 5 2

AB AC BC

DE DC EC

So, ΔABC ~ ΔDEC by the SSS Similarity Theorem.

Example 2: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

b)

Since the lengths of the sides that include M are proportional, ΔMNP ~ ΔMRS by the SAS Similarity Theorem.

By the Reflexive Property, M M.

12 12 2 10 10 2or and or .

12 18 30 5 10 15 25 5

MP MN

MS MR

Example 3:

a)If ΔRST and ΔXYZ are two triangles such that which of the

following would be sufficient to prove that the triangles are similar?

a) b)

c) R S d)

2,

3

RS

XY

RT ST

XZ YZ

RS RT ST

XY XZ YZ

RS XY

RT XZ

Example 3:

b) Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?

a)

b) mA = 2mD

c)

d)

4

3

AC

DC

AC BC

DC EC

5

4

BC

DC

Example 4:

a) Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT.

||RS UT

Since || , and because

they are alternate interior angles. By AA Similarity, .

Using the definition of similar polygons, .

RS UT SRQ UTQ RSQ TUQ

RSQ TUQ

RS RQ

TU TQ

4 3Substitution

10 2 104 2 10 10 3 Cross Products Property

8 40 10 30 Distributive Property

10 2 Subtract 8 and 30 from each side

5 Divide each side by 2

RS RQ

TU TQ

x

xx x

x x

x x

x

RQ = 8; QT = 20

Example 4:

b) Given , AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.

||AB DE

Since || , and because

they are alternate interior angles. By AA Similarity, .

Using the definition of similar polygons, .

AB DE BAC DEC ABC EDC

ABC EDC

AB AC

DE EC

38.5 3 8Substitution

11 238.5 2 11 3 8 Cross Products Property

38.5 77 33 88 Distributive Property

5.5 11 Subtract 33 and 77 from each side

2 Divide each side by 5.5

AB AC

DE ECx

xx x

x x

x x

x

AC = 14

Example 5:

a) Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?

In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So the following proportion can be written.

height of Sears Tower (ft) height of light pole (ft)

Sears Tower shadow length (ft) light pole shadow length (ft)

Substitute the known values and let x be the height of the Sears Tower.

12Substitution

242 22 242 12 Cross Products Property

2 2904 Simplify

1452 Divide each side by 2

x

x

x

x

The Sears Tower is 1452 feet tall.

Example 5:

b) On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?

In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So the following proportion can be written.

height of Lighthouse (ft) height of Jennie (ft)

Lighthouse shadow length (ft) Jennie's shadow length (ft)

Substitute the known values and let x be the height of the Cape Hatteras lighthouse.

5.5Substitution

53.5 1.51.5 53.5 5.5 Cross Products Property

1.5 294.25 Simplify

196.17 Divide each side by 1.5

x

x

x

x

The Cape Hatteras lighthouse is about 196 feet tall.