Download Lesson 4.6

4.6
Isosceles and Equilateral
CCSS
Content Standards
G.CO.10 Prove theorems about triangles.
G.CO.12 Make formal geometric constructions with a
variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding,
dynamic geometric software, etc.).
Mathematical Practices
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the
reasoning of others.
Then/Now
You identified isosceles and equilateral triangles.
• Use properties of isosceles triangles.
• Use properties of equilateral triangles.
Vocabulary
• legs of an isosceles triangle
• vertex angle
• base angles
Concept
Example
1
Congruent Segments and Angles
A. Name two unmarked congruent angles.
___
BCA is opposite___
BA and A
is opposite BC, so BCA 
A.
Answer: BCA and A
Example 1
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
Example 1a
A. Which statement correctly
names two congruent angles?
A. PJM  PMJ
B. JMK  JKM
C. KJP  JKP
D. PML  PLK
Example 1b
B. Which statement correctly names
two congruent segments?
A. JP  PL
B. PM  PJ
C. JK  MK
D. PM  PK
Concept
Example 2
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.
Answer:
mR = 60
.
.
Example 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
Example 2a
A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°
Example 2b
B. Find TS.
A. 1.5
B. 3.5
C. 4
D. 7
Example
3
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°.
Example 3
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
6y + 3 = 8y – 5
Definition of equilateral triangle
Substitution
3 = 2y – 5
Subtract 6y from each side.
8 = 2y
Add 5 to each side.
Example 3
Find Missing Values
4 =y
Answer: x = 17, y = 4
Divide each side by 2.
Example 3
Find the value of each variable.
A. x = 20, y = 8
B. x = 20, y = 7
C. x = 30, y = 8
D. x = 30, y = 7
Example 4
Apply Triangle Congruence
Given:
HEXAGO is a regular polygon.
ΔONG is equilateral, N is the midpoint of GE,
and EX || OG.
Prove:
ΔENX is equilateral.
___
Example
4 Congruence
Apply Triangle
Proof:
Statements
Reasons
1. HEXAGO is a regular polygon.
1. Given
2. ΔONG is equilateral.
2. Given
3. EX  XA  AG  GO  OH  HE
3. Definition of a regular
hexagon
4. N is the midpoint of GE.
4. Given
5. NG  NE
5. Midpoint Theorem
6. EX || OG
6. Given
Example
4 Congruence
Apply Triangle
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
Example 4
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
Example 4
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
A. Definition of isosceles triangle
B. Midpoint Theorem
C. CPCTC
D. Transitive Property