9-3 Proving Triangles Similar - Uplift Education · 9-3 Proving Triangles Similar Warm Up #8 1. ... Then set up ratios for each pair of corresponding ... Two pairs of proportional

9-3 Proving Triangles Similar

Do Now

Exit Ticket

Lesson Presentation


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


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.




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)


Vocabulary:

similarsimilar polygonssimilarity ratio


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:


In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.

Remember!


Figures that are similar (~) have the same shape but not necessarily the same size.


∆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°



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


Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations.

Helpful Hint


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.


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



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.



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.


Got It ? Solve With Your Partner

Problem 1 Using the AA ~ Postulate

39o

51o

AA ~ Postulate

44o

No. The included angle is not equal.



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



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.


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


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.

Documents

9-3 Proving Triangles Similar - Uplift Education · 9-3 Proving Triangles Similar Warm Up #8 1. ... Then set up ratios for each pair of corresponding ... Two pairs of proportional