Download Lesson 4-6

Five-Minute Check (over Lesson 4–5)
Then/Now
New Vocabulary
Theorems: Isosceles Triangle
Example 1: Congruent Segments and Angles
Corollaries: Equilateral Triangle
Example 2: Find Missing Measures
Example 3: Find Missing Values
Example 4: Real-World Example: Apply Triangle Congruence
Over Lesson 4–5
Refer to the figure. Complete the
congruence statement.
?
ΔWXY  Δ_____
by ASA.
A. ΔVXY
B. ΔVZY
C. ΔWYX
D. ΔZYW
A.
B.
C.
D.
A
B
C
D
Over Lesson 4–5
Refer to the figure. Complete the
congruence statement.
?
ΔWYZ  Δ_____
by AAS.
A. ΔVYX
B. ΔZYW
C. ΔZYV
D. ΔWYZ
A.
B.
C.
D.
A
B
C
D
Over Lesson 4–5
Refer to the figure. Complete the
congruence statement.
?
ΔVWZ  Δ_____
by SSS.
A. ΔWXZ
B. ΔVWX
C. ΔWVX
D. ΔYVX
A.
B.
C.
D.
A
B
C
D
Over Lesson 4–5
What congruence
statement is needed to
use AAS to prove
ΔCAT  ΔDOG?
C. A  G
0%
B
A
D. T  G
0%
A
B
C
0%
D
D
B. A  O
A.
B.
C.
0%
D.
C
A. C  D
You identified isosceles and equilateral
triangles. (Lesson 4–1)
• Use properties of isosceles triangles.
• Use properties of equilateral triangles.
• legs of an isosceles triangle
• vertex angle
• base angles
Congruent Segments and Angles
A. Name two unmarked congruent angles.
___
opposite
___BA
BCA is
and
A is opposite BC, so
BCA  A.
Answer: BCA and A
Congruent Segments and Angles
B. Name two unmarked congruent segments.
___
BC
is opposite D and
___
BD
is ___
opposite BCD, so
___
BC  BD.
Answer: BC  BD
A. Which statement correctly
names two congruent angles?
A. PJM  PMJ
B. JMK  JKM
D. PML  PLK
0%
B
A
0%
0%
C
C. KJP  JKP
A
B
C
D
0%
D
A.
B.
C.
D.
B. Which statement correctly
names two congruent segments?
A. JP  PL
B. PM  PJ
D. PM  PK
0%
B
A
0%
0%
C
C. JK  MK
A
B
C
D
0%
D
A.
B.
C.
D.
Find Missing Measures
A. Find mR.
Since QP = QR, QP  QR. By the
Isosceles Triangle Theorem, base
angles P and R are congruent, so
mP = mR . Use the Triangle Sum
Theorem to write and solve an equation
to find mR.
Triangle Sum Theorem
mQ = 60, mP = mR
Answer: mR = 60
Simplify.
Subtract 60 from each side.
Divide each side by 2.
Find Missing Measures
B. Find PR.
Since all three angles measure 60, the
triangle is equiangular. Because an
equiangular triangle is also equilateral,
QP = QR = PR. Since QP = 5, PR = 5
by substitution.
Answer: PR = 5 cm
A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°
0%
D
0%
C
0%
B
A
0%
A.
B.
C.
D.
A
B
C
D
B. Find TS.
A. 1.5
B. 3.5
C. 4
D. 7
0%
D
0%
C
0%
B
A
0%
A.
B.
C.
D.
A
B
C
D
Find Missing Values
ALGEBRA Find the value of each variable.
Since E = F, DE  FE by the Converse of the
Isosceles Triangle Theorem. DF  FE, so all of the
sides of the triangle are congruent. The triangle is
equilateral. Each angle of an equilateral triangle
measures 60°.
Find Missing Values
mDFE = 60
4x – 8 = 60
4x = 68
x = 17
Definition of equilateral triangle
Substitution
Add 8 to each side.
Divide each side by 4.
The triangle is equilateral, so all the sides are
congruent, and the lengths of all of the sides are equal.
DF = FE
Definition of equilateral triangle
6y + 3 = 8y – 5 Substitution
3 = 2y – 5 Subtract 6y from each side.
8 = 2y
Add 5 to each side.
Find Missing Values
4 =y
Answer: x = 17, y = 4
Divide each side by 2.
Find the value of each variable.
A. x = 20, y = 8
B. x = 20, y = 7
0%
B
A
0%
A
B
C
0%
D
D
D. x = 30, y = 7
A.
B.
C.
0%
D.
C
C. x = 30, y = 8
Apply Triangle Congruence
NATURE Many geometric figures
can be found in nature. Some
honeycombs are shaped like a
regular hexagon. That is, each of
the six sides and interior angle
measures are the same.
Given: HEXAGO is a regular polygon.
___
ΔONG is equilateral, N is the midpoint of GE,
and EX || OG.
Prove: ΔENX is equilateral.
Apply Triangle Congruence
Proof:
Statements
Reasons
1. HEXAGO is a regular polygon.
1. Given
2. ΔONG is equilateral.
2. Given
3. EX  XA  AG  GO  OH  HE
4. N is the midpoint of GE
3. Definition of a regular
hexagon
4. Given
5. NG  NE
5. Midpoint Theorem
6. EX || OG
6. Given
Apply Triangle Congruence
Proof:
Statements
7. NEX  NGO
8. ΔONG  ΔENX
Reasons
7. Alternate Exterior
Angles Theorem
8. SAS
9. OG  NO  GN
9. Definition of
Equilateral Triangle
10. NO  NX, GN  EN
10. CPCTC
11. XE  NX  EN
11. Substitution
12. ΔENX is equilateral.
12. Definition of
Equilateral Triangle
Given: HEXAGO is a regular hexagon.
NHE  HEN  NAG  AGN
___ ___ ___ ___
Prove: HN  EN  AN  GN
Proof:
Statements
Reasons
1. HEXAGO is a regular hexagon.
1. Given
2. NHE  HEN  NAG  AGN 2. Given
3. HE  EX  XA  AG  GO  OH
3. Definition of regular
hexagon
4. ΔHNE ΔANG
4. ASA
Proof:
Statements
Reasons
5. HN  AN, EN  NG
?
5. ___________
6. HN  EN, AN  GN
6. Converse of Isosceles
Triangle Theorem
7. HN  EN  AN  GN
7. Substitution
0%
B
D. Transitive Property
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A
C. CPCTC
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A
B
C
D
0%
D
B. Midpoint Theorem
C
A. Definition of isosceles triangle
A.
B.
C.
D.