Download Triangle Similarity

Geometry Solution Manual | Reference Guide Unit 8 | Triangle Similarity
Triangle Similarity
Solutions
1. Congruent
2. By the Polygon Similarity Postulate, in similar
polygons—including similar triangles—all pairs
of corresponding angles are congruent and all
pairs of corresponding sides are proportional.
The Triangle Sum Theorem states that the sum of
the measures of the three angles of any triangle
is 180°. If two pairs of corresponding angles of
a triangle are congruent, then the third pair of
angles must also be congruent—and the sides
will be proportional. Those reasons are enough to
prove that the two triangles are similar.
3. △ ABC ∼ △ DEF ∼ △ JKL (These can be
written as pairs of similar triangles or as shown.)
4. A. m∠ L = 180° − (80° + 50°) = 50°;
B.
C.
D.
the triangles are not similar because
corresponding angles are not congruent.
m∠ B = 180° − (60° + 40°) = 80°
m∠ B = m∠ R = 80°
m∠ C = m∠ S = 80°
△ RST ∼ △ BCA because of the AA
Similarity Postulate. If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
m∠ Q = m∠ R = 80°
m∠ P = m∠ S = 80°
△ RST ∼ △QPR because of the AA
Similarity Postulate. If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
There is not enough information given. At
least one more angle is needed.
© 2009 K12 Inc. All rights reserved.
Copying or distributing without K12’s written consent is prohibited.
5. A. m∠ D = 180° − (50° + 40°) = 90°
m∠ D = m∠ L = 90°
m∠ F = m∠ M = 50°
△ JML ∼ △ DEF because of the AA
B.
C.
D.
Similarity Postulate. If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
m∠C = 180° − (40° + 40°) = 100°;
the triangles are not similar because
corresponding angles are not congruent.
m∠W = 180° − (40° + 45°) = 95°;
the triangles are not similar because
corresponding angles are not congruent.
m∠ F = 180° − (50 + 90)° = 40°
m∠ F = m∠ J = 40°
m∠ M = m∠ D = 50°
△ JML ∼ △ FDE because of the AA
Similarity Postulate. If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
1 of 4
Geometry Solution Manual | Reference Guide Unit 8 | Triangle Similarity
6.
___
___
Given: KG​
​  and ​JH​ intersect at point I
m∠ J = 45°
m∠ H = 45°
IJ = 6 mm
K
45°
I
G
Note: Figure is not drawn to scale.
Prove: GH = 6 mm
45°
J
Statement
a. m∠ J = 45°
m∠ H = 45°
b. m∠ J = m∠ H
Reasons
a. Given
c. KG​
​  and ​JK​ intersect at point I
c. Given
d. ∠ KIJ and ∠ GIH are vertical angles
d. Definition of vertical angles
e. ∠ KIJ ≅ ∠ GIH
e. Vertical angles are congruent
f. △GHI ∼ △ KJI
f. AA Similarity Postulate
IG
IH ___
___
KJ ​ = ​ IJ ​ = ​ IK  ​
GH ___
8
____
h. ​   ​ = ​    ​ 
16
12
i. GH (16) = (8)(12)
96
GH = ​ ___
16 ​ = 6 mm
g. Polygon Similarity Postulate
___
b. Substitution Property of Equality
___
GH
____
g. ​ 
7.
H
h. Substitution Property of Equality
i. Means-Extremes Product Property
Given: m∠Q = 30°
m∠ 60°
___
___S =
TU ​
​  ∥ ​RS ​ 
Q
Prove: △QRS ∼ △QTU
30°
U
T
60°
R
Statement
a. m∠Q = 30°
m∠ S = 60°
Reasons
a. Given
b. ​TU ​ || ​RS ​ 
b. Given
___
___
___
c. ​___
QS​ is a transversal
___
​TU ​ and ​RS ​ 
that intersects
d. m∠TUQ = 60°
e. ∠Q ≅ ∠Q
f. ∠ S ≅ ∠ TUQ
g. △QRS ∼ △QTU
© 2009 K12 Inc. All rights reserved.
Copying or distributing without K12’s written consent is prohibited.
S
c. Definition of Transversal
d. If two parallel lines are cut by a
transversal, the corresponding angles are
congruent
e. Reflexive Property of Equality
f. Transitive Property of Equality
g. AA Similarity Postulate
2 of 4
Geometry Solution Manual | Reference Guide Unit 8 | Triangle Similarity
8. Proportional
15. A. AA Similarity Postulate
9. Included
10. SAS Similarity Postulate
11. A. SSS Similarity Theorem
B. ∠ X ≅ ∠ D because by the Polygon Similarity
Postulate, in similar polygons, all pairs of
corresponding angles are congruent.
12.
AA Similarity Postulate
m∠ H = 180° − (90° + 26°) = 64°
m∠ K = 180° − (90° + 64°) = 26°
m∠ H = m∠ L and m∠G = m∠ K
B. One pair of congruent corresponding angles is
∠ K and ∠ R and the other pair is ∠ L and ∠ S.
C. △ JKL ∼ △QRS
16. A. SAS Similarity Theorem
AC ___
___
  ​= ​ AB   ​
B. ​ 
QS
QB
___
​ 3.2 ​ = __
​ 6 ​ 
1.6 3
2=2
∠ A and ∠Q are corresponding and are
congruent.
C. △CAB ∼ △ SQB
13. SSS Similarity Postulate
14. SAS Similarity Theorem
17. A. SSS Similarity Theorem
QP NO
​   ​ 
​ ___  ​= ___
RQ
SR
12
____
____
​    ​ = ​  8   ​ 
21.6 14.4
4   ​ = ___
​ ___
​  4   ​ 
7.2 7.2
Two pairs of corresponding sides are
proportional. The angles included between the
two pairs of sides are each 55°. Thus the included
corresponding angles are congruent.
© 2009 K12 Inc. All rights reserved.
Copying or distributing without K12’s written consent is prohibited.
YX  ​= ____
B. ___
​ WY ​ = ​ ___
​ XW  ​
PR RQ QP
___
​ 0.5 ​ = ___
​ 0.5 ​  = ___
​ 0.5 ​ 
2
2
2
C. △WXY ∼ △ PQR
3 of 4
Geometry Solution Manual | Reference Guide Unit 8 | Triangle Similarity
18.
___
D
Given: Point B is the midpoint
of EC​
​  and
___
the midpoint of AD​
​  
Prove: △
AEB ∼ △ DCB C
B
E
A
Statement
___
___
a. Point B is the midpoint of EC​
​  and AD​
​  
Reasons
a. Given
___
___
b. ___
​  ≅ BC​
EB​
​  
___
​  
​AB​ ≅ BD​
b. Definition of midpoint
c. ∠ EBA and ∠CBD are vertical angles
c. Definition of vertical angles
d. ∠ EBA ≅ ∠CBD
d. Vertical angles are congruent
e. △ AEB ∼ △ DCB
e. SAS Similarity Theorem
© 2009 K12 Inc. All rights reserved.
Copying or distributing without K12’s written consent is prohibited.
4 of 4