Download geometry module 2 lesson 14-17 similar triangles

GEOMETRY
MODULE 2 LESSON 14-17
SIMILAR TRIANGLES
Similar Triangles: Two triangles are similar if and only if the corresponding sides are in proportion
and the corresponding angles are congruent. Notation: ~
Similarity of triangles are…

Reflexive: A triangle is similar to itself. ∆𝐴𝐵𝐶 ~ ∆𝐴𝐵𝐶

Symmetric: If ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹, then ∆𝐷𝐸𝐹~∆𝐴𝐵𝐶.

Transitive: If ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹 and ∆𝐷𝐸𝐹~∆𝐺𝐻𝐼, then ∆𝐴𝐵𝐶~∆𝐺𝐻𝐼.
CRITERION FOR TWO TRIANGLES TO BE SIMILAR

Angle-Angle (AA)

Side-Side-Side (SSS)

Side-Angle-Side (SAS)
AA Criterion
If the two angles of one triangle are congruent to two angles of another triangle, then the triangles
are similar.
∠𝐴 ≅ ∠𝐷 and ∠𝐵 ≅ ∠𝐸, so ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹.
SSS Criterion
If the measures of the corresponding sides of two triangles are proportional, then the triangles are
similar.
𝐴𝐵
𝐷𝐸
𝐵𝐶
𝐴𝐶
= 𝐸𝐹 = 𝐷𝐹 , so ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹.
SAS Criterion
MOD2 L14-17
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If the measures of two sides of a triangle are proportional to the measures of two corresponding
sides of another triangle and the included angles are congruent, then the triangles are similar.
𝐴𝐵
𝐷𝐸
𝐴𝐶
= 𝐷𝐹 and ∠𝐴 ≅ ∠𝐷, so ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹.
PRACTICE
Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing
angle and side-length measures.
1.
x
y
The missing angles are calculated using Sum of the Angles in a Triangle= 180°. We find that there
are indeed two pairs of equal corresponding angles. By AA criterion, the triangles are similar.
To find the missing sides requires a bit more work. First set up the proportions.
6
7
𝑥
=
=
8 28 𝑦
3
𝑥
0.75 =
𝑦
Since x is opposite the smallest angle, we need to choose a value for x that is smaller than 6 and 7.
5
Let 𝑥 = 5. Therefore, 𝑦 = 0.75 =
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3
2
2.
4
3.14
𝑥
=
=
12
𝑦
16.5
1 3.14
1
𝑥
=
𝑎𝑛𝑑
=
3
𝑦
3 16.5
𝑦 = 3(3.14) = 9.42
𝑥=
16.5
3
= 5.5
Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity
statement.
1.
A.
B.
𝐴𝐵
𝐴𝐶
𝐵𝐶
= ′ ′=
′
′
𝐴𝐵
𝐴𝐶
𝐵′𝐶′
2.82 5.4
7
=
=
1.41 2.7 3.5
2 = 2 = 2
∆𝐴𝐵𝐶~∆𝐴′ 𝐵 ′ 𝐶 ′ 𝑏𝑦 𝑆𝑆𝑆 𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛
𝐴𝐵 𝐴𝐶
𝐵𝐶
=
=
𝐸𝐹 𝐷𝐹 𝐷𝐸′
1
5.83
6.4
=
=
0.68 2.13 2.42
1.47 ≠ 2.74 ≠ 2.64
The triangles are not similar.
MOD2 L14-17
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2.
A.
𝐴𝐵 𝐶𝐵
=
𝐷𝐸 𝐸𝐹
3
3.13
=
1.5 1.565
2 = 2
∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 𝑏𝑦 𝑆𝐴𝑆 𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛
B.
𝐴𝐸 𝐵𝐸
=
𝐷𝐸 𝐸𝐶
2.24 3.16
=
4.48 6.32
.5 = .5
∆𝐴𝐸𝐵~∆𝐷𝐸𝐶 𝑏𝑦 𝑆𝐴𝑆 𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛
A
Brian is photographing the Washington Monument and
wonders how tall the monument is. Brian places his 5ft.
camera tripod approximately 100yd. from the base of the
monument. Lying on the ground, he visually aligns the top
of his tripod with the monument and marks his location on
the ground approximately 2ft. 9in from the center of his
tripod. Use Brian’s measurements to approximate the height
B
of the Washington Monument.
Currently, the given values are presented in three different
units. We must convert to the same unit. We will convert to
feet. 100𝑦𝑑 = 300𝑓𝑡 and 2𝑓𝑡 9𝑖𝑛 = 2.75𝑓𝑡
E
D
𝐴𝐸 𝐸𝐶
=
𝐵𝐷 𝐶𝐷
𝐴𝐸 302.75
=
5
2.75
𝐴𝐸 =
(302.75)(5)
= 550.5 𝑓𝑡
2.75
MOD2 L14-17
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C
ON YOUR OWN
1. Catarina’s boat has come untied and floated
A
away on the lake. She is standing atop a cliff
that is 35ft above the water in a lake. If she
stands 10ft. from the edge of the cliff, she
C
B
can visually align the top of the cliff with the
water at the back of her boat. Her eye level
is 5 ½ ft. above the ground. Approximately
how far out from the cliff is Catarina’s boat?
D
E
𝐴𝐸 𝐵𝐶
=
𝐶𝐷 𝐷𝐸
5.5 10
=
35 𝐷𝐸
𝐴𝐸 =
350
= 63.6 𝑓𝑡
5.5
MOD2 L14-17
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̅̅̅̅ ⊥ 𝑉𝑊
̅̅̅̅̅ and 𝑊𝑌
̅̅̅̅̅ ⊥ 𝑈𝑉
̅̅̅̅.
Given the diagram to the right, 𝑈𝑋
Show that ∆𝑈𝑋𝑉~∆𝑊𝑌𝑉.
STEP
JUSTIFICATION
̅̅̅̅ ⊥ 𝑉𝑊
̅̅̅̅̅ and 𝑊𝑌
̅̅̅̅̅ ⊥ 𝑈𝑉
̅̅̅̅
𝑈𝑋
Given
∠𝑈𝑋𝑉 ≅ ∠𝑊𝑌𝑉
Right angles = 90° so angles equal to each other
∠𝑉 ≅ ∠𝑉
Both triangles share ∠𝑉 , so Reflexive
∆𝑈𝑋𝑉~∆𝑊𝑌𝑉
AA Criterion
Are the triangles shown similar?
Explain. If the triangles are similar,
write the similarity statement.
Yes, the triangles are similar. ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸 by AA because ∠𝐴𝐷𝐸 ≅ ∠𝐴𝐵𝐶, and both
triangles share ∠𝐴.
MOD2 L14-17
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