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4-5
Isosceles and Equilateral Triangles
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Exit Ticket
4-5
Isosceles and Equilateral Triangles
Warm Up # 3
1. Find each angle measure.
60°; 60°; 60°
True or False. If false explain.
2. Every equilateral triangle is isosceles.
True
3. Every isosceles triangle is equilateral.
False; an isosceles triangle can have
only two congruent sides.
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Isosceles and Equilateral Triangles
Knowledge: Justify Mathematical Argument (1)(G)
A builder using the truss shown at the right claims that
ACB will have the same measure as ADB. 𝑨𝑪 and 𝑨𝑫
represent identical beams, and 𝑨𝑩 bisects CAD. Is the
builder correct? Justify your answer.
Yes. The builder is correct.
It is given that 𝐴𝐶 ≌ 𝐴𝐷 and by definition
of angle bisectors, CAB ≌ DAB.
By the Reflexive Prop. of ≌, 𝐴𝐵 ≌ 𝐴𝐵.
Thus, ∆ACB ≌ ∆ADB by SAS Postulate.
ACB ≌ ADB because of CPCTC.
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Isosceles and Equilateral Triangles
Knowledge: Making a Conjecture
A. Construct congruent segments to make a conjecture
about the angles opposite the congruent sides in an
isosceles triangle.
Step 1: Construct an isosceles ∆ABC on your paper, with
𝐴𝐶 ≅ 𝐵𝐶.
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Isosceles and Equilateral Triangles
Know: Making a Conjecture
Construct congruent segments to make a conjecture
about the angles opposite the congruent sides in an
isosceles triangle.
Step 2: Fold the paper so that the two congruent sides fit exactly one
on top of the other. Create the paper. Notice that A and B appear to be
congruent.
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Isosceles and Equilateral Triangles
Communicate: Connect Mathematical Ideas (1)(F)
Think: How can folding a piece of paper help you
tell if two angles are congruent?
When folding the paper, congruent angles will fit
exactly one on top of the other.
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Isosceles and Equilateral Triangles
Knowledge: Making a Conjecture
Write a conjecture that you observed for the angles
opposite the congruent sides in an isosceles triangle.
Angles opposite the congruent sides in an isosceles
triangle are congruent.
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Isosceles and Equilateral Triangles
Knowledge: Making a Conjecture
Write a conjecture that you observed for the sides
opposite the congruent angles in an isosceles triangle.
Sides opposite the congruent angles in an isosceles
triangle are congruent.
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Isosceles and Equilateral Triangles
Connect to Math
By the end of today’s lesson,
SWBAT
1. Prove theorems about isosceles and equilateral triangles.
2. Apply properties of isosceles and equilateral triangles.
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Isosceles and Equilateral Triangles
Vocabulary
legs of an isosceles triangle
vertex angle
base
base angles
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Isosceles and Equilateral Triangles
Recall that an isosceles triangle has at least two
congruent sides. The congruent sides are called
the legs. The vertex angle is the angle formed
by the legs. The side opposite the vertex angle is
called the base, and the base angles are the
two angles that have the base as a side.
3 is the vertex angle.
1 and 2 are the base angles.
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Isosceles and Equilateral Triangles
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Isosceles and Equilateral Triangles
Example 1: Proving the Isosceles Triangle Theorem
Begin with isosceles ∆XYZ with 𝑿𝒀 ≅ 𝑿𝒁. Draw 𝑿𝑩, the bisector
of vertex angle YXZ.
Given: 𝑋𝑌 ≅ 𝑋𝑍, 𝑋𝐵 bisects YXZ
Prove: Y ≌ Z
Statements
1. 𝑋𝑌  𝑋𝑍 ; 𝑋𝐵 bisects YXZ
2. 1  2
3. 𝑋𝐵  𝑋𝐵
4. ∆XYB  ∆XZB
5. Y ≌ Z
Reasons
1. Given
2. Definition of angle bisector
3. Reflex. Prop. of 
4. SAS Postulate Steps 1, 2, 3
5. CPCTC
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Isosceles and Equilateral Triangles
Example 2: Proving the Isosceles Triangle Theorem
A builder using the truss shown at the right claims that
ACB will have the same measure as ADB. 𝑨𝑪 and 𝑨𝑫
represent identical beams, and 𝑨𝑩 bisects CAD. Is the
builder correct? Justify your answer.
Yes. The builder is correct.
It is given that 𝐴𝐶 ≌ 𝐴𝐷 by the Isosceles
Triangle Theorem.
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Isosceles and Equilateral Triangles
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Isosceles and Equilateral Triangles
Example 3:
Using the Isosceles Triangle Theorem and its Converse
A. Is 𝑨𝑩 congruent to 𝑪𝑩 ? Explain.
Yes. Since C ≌ A, 𝐴𝐵 ≅ 𝐶𝐵 by the
Converse of the Isosceles Triangle Theorem.
B. Is A congruent to DEA ? Explain.
Yes. Since 𝐴𝐷 ≅ 𝐸𝐷, A ≌ DEA by the
Isosceles Triangle Theorem.
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Isosceles and Equilateral Triangles
Reading Math
The Isosceles Triangle Theorem is
sometimes stated as “Base angles of
an isosceles triangle are congruent.”
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Isosceles and Equilateral Triangles
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Isosceles and Equilateral Triangles
Example 4: Using Algebra
What is the value of x ?
Since 𝐴𝐵 ≅ 𝐶𝐵, ∆ABD is isosceles ∆. By
the Isosceles ∆ Theorem A ≌ C.
mC = 54o
Since 𝐵𝐷 bisects ABC, you know by Theorem 4-5 that
𝐵𝐷 ⊥ 𝐴𝐶. So, BDC = 90o.
mC + mBDC + mDBC = 180o
54 + 90 + x = 180o
x = 36o
∆ Sum Theorem.
Substitute.
Subtract 144 from each side.
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Isosceles and Equilateral Triangles
Example 5: Complete each Statement.
Explain why it is true.
a. 𝑽𝑻 ≅ ________
𝑉𝑋
Converse of Isosceles ∆ Theorem
𝑈𝑊 ≌ 𝒀𝑿 Converse of Isosceles ∆ Thrm.
b. 𝑼𝑻 ≅ ________
𝑉𝑌
c. 𝑽𝑼 ≅ ________
Converse of Isosceles ∆ Thrm.
and Segment Addition Post.
𝑉𝑈𝑌 Isosceles ∆ Theorem
d. 𝑽𝒀𝑼 ≅ ________
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Isosceles and Equilateral Triangles
The following corollary and its
converse show the connection
between equilateral triangles
and equiangular triangles.
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Isosceles and Equilateral Triangles
Equilateral Triangle
Equiangular Triangle
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Isosceles and Equilateral Triangles
Example 6: Using Algebra
B
A. What is the value of x ?
40o
Because x is the measure of an angle
in an equilateral triangle, x = 60o.
F
D
yo
xo
A
C
E
G
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Isosceles and Equilateral Triangles
Example 6: Using Algebra
B
B. What is the value of y ?
40o
F
mDCE + mDEC + mEDC = 180.
∆ Sum Theorem.
Substitute.
D
60 + 70 + y = 180
Subtract 130 from each side.
yo
y = 50
xo
A
C
E
G
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Isosceles and Equilateral Triangles
Example 7: Using Algebra
A. What is the value of x ?
x + 2y = 180o ∆ Sum Theorem.
x + 2(70) = 180 Substitute.
x = 40o Subtract 140 from each side.
B. What is the value of y ?
It is given that the triangle is an isosceles ∆. Thus, the base
angles are congruent. Since 110o and the base angle to y
are linear pair. Hence, y = 70o by Linear Pair Postulate.
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Isosceles and Equilateral Triangles
Example 8: Using Algebra
The vertex angle of an isosceles triangle
measures (a + 15)°, and one of the base angles
measures 7a°. Find a and each angle measure.
a + 15 + 7a + 7a = 180o ∆ Sum Theorem.
15a + 15 = 180
15a = 165
a = 11
(a + 15)°
Combined Like Terms
Subtract 15 from each side.
Subtract 15 from each side.
Therefore, each angle measure is 26°; 77°; 77°
7a°
7a°
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Isosceles and Equilateral Triangles
Exit Ticket:
Find each angle measure.
1. mR
2. mP
Find each value.
3. x
5. x
4. y