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9-3 Proving Triangles Similar
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9-3 Proving Triangles Similar
Warm Up #81. The ratio of the angle measures in a triangle is 1:5:6. What is
the measure of each angle?
Solve each proportion.
2. 3.
4. Given that 14a = 35b, find the ratio of a to b in simplest form.
5. An apartment building is 90 ft tall and 55 ft wide. If a scale
model of this building is 11 in. wide, how tall is the scale model
of the building?
15°, 75°, 90°
x = 3 x = 7 or –7
18 in.
𝑎
𝑏=
5
2
9-3 Proving Triangles Similar
The Lord of the Rings movies transport viewers to the fantasy world of Middle Earth. Many scenes feature vast fortresses, sprawling cities, and bottomless mines. To film these images, the moviemakers used ratios to help them build highly detailed miniature models.
9-3 Proving Triangles Similar
9-3 Proving Triangles Similar
9-3 Proving Triangles Similar
SWBAT
Prove certain triangles are similar by using AA, SSS, and SAS.
Use triangle similarity to solve problems.
By the end of today’s lesson,
Connect to Mathematical Ideas (1)(F)
9-3 Proving Triangles Similar
Vocabulary:
similarsimilar polygonssimilarity ratio
9-3 Proving Triangles Similar
A ratio compares two numbers by division. The ratio
of two numbers a and b can be written as a to b, a:b,
or , where b ≠ 0. For example, the ratios 1 to 2,
1:2, and all represent the same comparison.
Prior Knowledge:
9-3 Proving Triangles Similar
In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.
Remember!
9-3 Proving Triangles Similar
Figures that are similar (~) have the same shape but not necessarily the same size.
9-3 Proving Triangles Similar
∆PQR and ∆STW
Example 1: Identifying Similar Polygons
Determine whether the triangles are similar. If so, write the similarity ratio and a similarity statement.
Step 1 Identify pairs of congruent angles.
P R and S W isos. ∆
Step 2 Compare corresponding angles.
Since no pairs of angles are congruent, the triangles are not similar.
mW = mS = 62°
mT = 180° – 2(62°) = 56°
9-3 Proving Triangles Similar
9-3 Proving Triangles Similar
Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique.
JL
LK=
JM
MK
20
x=
15
27.75
9-3 Proving Triangles Similar
Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations.
Helpful Hint
9-3 Proving Triangles Similar
Example 2: Measurement Application
Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?
Step 1 Convert the measurements to inches.
Step 2 Find similar triangles.
Because the sun’s rays are parallel, A F. Therefore ∆ABC ~ ∆FGH by AA ~.
AB = 7 ft 8 in. = (7 12) in. + 8 in. = 92 in.
BC = 5 ft 9 in. = (5 12) in. + 9 in. = 69 in.
FG = 38 ft 4 in. = (38 12) in. + 4 in. = 460 in.
9-3 Proving Triangles Similar
Example 2: Measurement Application
Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?
Step 3 Find h.
Therefore, the height h of the pole
is 345 inches, or 28 feet 9 inches.
Corr. sides are proportional.
Substitute 69 for BC, h for GH, 92 for AB, and 460 for FG.
Cross Products Prop.92h = 69 460
Divide both sides by 92.h = 345
9-3 Proving Triangles Similar
9-3 Proving Triangles Similar
Example 3: Verifying Triangle Similarity
K
L
P K
M
N
8
12 12 + 3 = 15
K is the included angle between two known sides in each triangle.
Since K ≌ K by Reflexive Property of Congruence.
𝐾𝐿
𝐾𝑀=
8
10=
4
5
𝐾𝑃
𝐾𝑁=
12
15=
4
5and
Thus, ∆KLP ~ ∆KMN by the SAS ~ Theorem.
9-3 Proving Triangles Similar
9-3 Proving Triangles Similar
Example 4: Verifying Triangle Similarity
Are the triangles similar? If so, write a similarity statement for the triangles.
Use the side lengths to identify corresponding sides. Then set up ratios for each pair of corresponding sides.
All three ratios are equal, so corresponding sides are proportional. Thus, ∆STU ~ ∆XVW by the SSS ~ Thrm.
9-3 Proving Triangles Similar
Got It ? Solve With Your Partner
Problem 1 Using the AA ~ Postulate
39o
51o
AA ~ Postulate
44o
No. The included angle is not equal.
9-3 Proving Triangles Similar
Got It ? Solve With Your Partner
Problem 2 Verifying Triangle Similarity
Are the triangles similar? If so, write a similarity statement for the triangles and explain how you know the triangles are similar?
Yes, by SSS ~
𝐴𝐶
𝐸𝐺=
𝐵𝐶
𝐹𝐺=
6
8=
3
4
𝐴𝐵
𝐸𝐹=
9
12=
3
4Yes, by SAS ~
𝐴𝑊
𝐴𝐸=
6
12=
1
2
𝐴𝐿
𝐴𝐶=
8
16=
1
2and
9-3 Proving Triangles Similar
Got It ? Solve With Your Partner
Problem 3 Explain why the triangles are similar. Then find the distance represented by x.
a. b.
A pair of congruent angles are given and the two right angles are ≅, so the triangles are similar by AA ~; x = 13.75 ft. or 13 ft. 9 in.
A pair of congruent angles are vertical angles and a pair of ≅ right angles, so the triangles are similar by AA ~; x = 80 ft.
9-3 Proving Triangles Similar
Closure: Communicate Mathematical Ideas (1)(G)
What is the minimum number of conditions necessary to prove two triangles similar? What are the conditions?
What other sets of conditions will prove that two triangles are similar?
Two pairs of proportional side lengths and the included angle, or three pairs of proportional side lengths.
Two pairs of congruent angles are enough to prove two triangles similar
9-3 Proving Triangles Similar
1. Explain why the triangles are similar and write a similarity
statement.
2. Explain why the triangles are similar, then find BE and CD.
Exit Ticket:
1. By the Isosc. ∆ Thm., A C, so by the def. of , mC = mA. Thus mC = 70° by subst. By the ∆ Sum Thm., mB = 40°. Apply the Isosc. ∆ Thm. and the ∆ Sum Thm. to ∆PQR. mR = mP = 70°. So by the def. of , A P, and C R. Therefore ∆ABC ~ ∆PQR by AA ~.
2. A A by the Reflex. Prop. of . Since BE || CD, ABE ACD by the Corr. s Post. Therefore ∆ABE ~ ∆ACD by AA ~. BE = 4 and CD = 10.