Page 127 - Math Course 3 (Book 2)

Similarity of Triangles

Since AB || DC, ∠BAC ≅ ∠DCE by the Alternate Now find RQ and QT.
Interior Angles Theorem. RQ = x + 3 QT = 2x + 10
= 5 + 3 or 8 = 2(5) + 10 or 20
Vertical angles are congruent, so ∠BAE ≅ ∠DEC.

Therefore, by the AA Similarity
Answer Answer RQ = 8; QT = 20
Theorem, ΔABE ~ ΔCDE.

Parts of Similar Triangles Indirect Measurement

Example Example

ALGEBRA INDIRECT MEASUREMENT
Josh wanted to measure the height of the Sears
Given RS || UT, RS = 4, RQ = x + 3, QT = 2x + 10,
Tower in Chicago. He used a 12-foot light pole and
UT = 10, find RQ and QT. measured its shadow at 1 P.M. The length of the
shadow was 2 feet. Then he measured the length
of the Sears Tower’s shadow and it was 242 feet at
R that time. What is the height of the Sears Tower?

x + 3 4

S
Q drawing
not to
U scale
x ft
2x + 10

10 12ft

T 2ft
242 ft
Since RS || UT, ∠SRQ ≅ ∠UTQ and ∠RSQ ≅ ∠TUQ
because they are alternate interior angles. By AA Since the sun’s rays form similar triangles, the
Similarity, ΔRSQ ~ ΔTUQ. Using the definition of following proportion can be written.
similar polygons, RS = RQ
TU TQ height of Sear’s Tower (ft) = height of the light pole (ft)
Sear’s Tower shadow length(ft) light pole shadow length (ft)
RS RQ Now substitute the known values and let x be the
TU = TQ height of the Sears Tower.
4 = x + 3 Substitution x 12
10 2x + 10 242 = 2 Substitution
4(2x + 10) = 10(x + 3) Cross products x • 2 = 242(12) Cross products

8x + 40 = 10x + 30 Distributive Property 2x = 2904 Simplify.
Subtract 8x and 30 from x = 1452 Divide each side by 2.
10 = 2x
each side. The Sears Tower is 1452 feet
Answer
5 = x Divide each side by 2. tall.

119