Download Reteach 7.3

Name ________________________________________ Date ___________________ Class __________________
LESSON
7-3
Reteach
Triangle Similarity: AA, SSS, and SAS
Angle-Angle (AA)
Similarity
Side-Side-Side (SSS)
Similarity
Side-Angle-Side (SAS)
Similarity
If two angles of one triangle
are congruent to two angles of
another triangle, then the
triangles are similar.
If the three sides of one
triangle are proportional to the
three corresponding sides of
another triangle, then the
triangles are similar.
If two sides of one triangle are
proportional to two sides of
another triangle and their
included angles are
congruent, then the triangles
are similar.
UABC ∼ UDEF
UABC ∼ UDEF
UABC ∼ UDEF
Explain how you know the triangles are similar, and write a similarity
statement.
1.
2.
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3. Verify that UABC ∼ UMNP.
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7-22
Holt McDougal Geometry
Name ________________________________________ Date ___________________ Class __________________
LESSON
7-3
Reteach
Triangle Similarity: AA, SSS, and SAS continued
You can use AA Similarity, SSS Similarity, and SAS Similarity to solve problems.
First, prove that the triangles are similar. Then use the properties of similarity to
find missing measures.
Explain why UADE ∼ UABC and then find BC.
Step 1
Prove that the triangles are similar.
∠A ≅ ∠A by the Reflexive Property of ≅.
AD 3 1
= =
AB 6 2
AE 2 1
= =
AC 4 2
Therefore, UADE ∼ UABC by SAS ∼.
Step 2
Find BC.
AD DE
=
AB BC
3 3.5
=
6 BC
Corresponding sides are proportional.
Substitute 3 for AD, 6 for AB, and 3.5 for DE.
3(BC) = 6(3.5)
Cross Products Property
3(BC) = 21
Simplify.
BC = 7
Divide both sides by 3.
Explain why the triangles are similar and then find each length.
4. GK
5. US
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-23
Holt McDougal Geometry
2. Possible answer: Every equilateral
triangle is also equiangular, so each angle
in both triangles measures 60°. Thus,
UTUV ∼ UWXY by AA ∼.
∠RQS. It is given that ∠H ≅ ∠Q. So by
the Angle Addition Postulate, ∠KHJ ≅
HK HJ 3
∠TQS.
=
= , so UKHJ ∼ UTQS
QT QS 2
by SAS∼. Because UKHJ ∼ UTQS,
JK HK 3
=
= . All the corresponding
ST QT 2
angles are congruent; all the
corresponding sides are proportional.
Thus, GHIJK ∼ PQRST by the definition
of similar polygons.
3. Possible answer: It is given that ∠JMN ≅
KL JL 4
∠L.
=
= . Thus, UJLK ∼ UJMN
MN JM 3
by SAS ∼.
PQ QR PR 3
4. Possible answer:
=
=
= .
UT TS US 5
Thus, UPQR ∼ UUTS by SSS ∼.
5. Possible answer: ∠C ≅ ∠C by the
Reflexive Property. ∠CGD and ∠F are
right angles, so they are congruent. Thus,
UCDG ∼ UCEF by AA ∼. DE = 9.75
Reteach
1. ∠Q ≅ ∠T by the Def. of ≅ ∠s . By the U
Sum Thm., m∠S = 39°and m∠U = 49°, so
∠S ≅ ∠V and ∠R ≅ ∠U. UQRS ∼ UTUV
by AA ∼.
2. ∠HJG ≅ ∠LJK by the Vert. ∠s Thm.
HJ GJ 2
=
= . UGHJ ∼ UKLJ by SAS ∼.
LJ KJ 3
AB BC CA 4
3.
=
=
= ; UABC ∼ UMNP by
MN NP PM 5
SSS ∼.
Practice C
1. Possible answer: UABC and UADB
share ∠A. They also each have a right
angle, so UABC ∼ UADB by AA ∼. They
2
have a similarity ratio of . UABC and
1
UBDC share ∠C. They also each have a
right angle, so UABC ∼ UBDC by AA ∼.
2 3
to 1.
They have a similarity ratio of
3
By the Transitive Property of Similarity,
UADB ∼ UBDC. They have a similarity
3
to 1.
ratio of
3
4. JK || FH , so ∠J ≅ ∠H, and ∠K ≅ ∠F by
the Alt. Int. ∠s Thm. UJKG ∼ UHFG by
1
AA ∼. GK = 7
3
5. It is given that ∠S ≅ ∠WVU. ∠U ≅ ∠U by
the Reflex. Prop. of ≅. UUVW ∼ UUST
by AA ∼. US = 39
2. 6 + 2 3
3. 3 +
3;3+3 3
4.
Challenge
13
3
1.
Statements
5. Possible answer: Draw diagonals
HK , HJ , QS, and QT . ∠G and ∠P are
right angles, so they are
GK GH 3
congruent.
=
= , so UGHK ∼
PT PQ 2
UPQT by SAS∼. It is given that ∠I ≅
HI
IJ
3
∠R.
=
= , so UHIJ ∼ UQRS by
QR RS 2
SAS∼. Because UGHK ∼ UPQT,
HQ 3
= and ∠GHK ≅ ∠PQT. Because
QT 2
HJ 3
UHIJ ∼ UQRS,
= and ∠IHJ ≅
QS 2
Reasons
∠PQR is a right angle. QS is
the altitude of UPQR drawn
from the right angle.
1. Given
QS ⊥ PR
2. Definition of
altitude
∠PSQ and ∠QSR are right
angles.
3. Definition of
perpendicular
m∠PSQ = m∠QSR =
m∠PQR = 90°
4. Definition of right
angle
∠PSQ ≅ ∠PQR; ∠QSR ≅
∠PQR
5. Definition of
congruent angles
∠P ≅ ∠P; ∠R ≅ ∠R
6. Reflexive
Property of
Congruence
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A5
Holt McDougal Geometry