4.6 Isosceles and Equilateral
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Transcript 4.6 Isosceles and Equilateral
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 mR.
Since QP = QR, QP QR. By the Isosceles
Triangle Theorem, base angles P and R are
congruent, so
mP = mR . Use the Triangle Sum
Theorem to write and solve an equation to
find mR.
Answer:
mR = 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 mT.
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
mDFE = 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