1. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
(For help, go to Lesson 3-4.)
1. Name the angle opposite AB.
2. Name the angle opposite BC.
3. Name the side opposite A.
4. Name the side opposite C.
5. Find the value of x.
By the Triangle Exterior Angle Theorem, x = = 105°.
Check Skills You’ll Need
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2. Using Congruent Triangles: CPCTC
GEOMETRY LESSON 4-4
1. What does “CPCTC” stand for?
Use the diagram for Exercises 2 and 3.
2. Tell how you would show ABM ACM.
3. Tell what other parts are congruent by CPCTC.
Use the diagram for Exercises 4 and 5.
4. Tell how you would show RUQ TUS.
5. Tell what other parts are congruent by CPCTC.
Corresponding parts of congruent triangles are congruent.
You are given two pairs of s, and AM AM by the Reflexive Prop., so ABM ACM by ASA.
AB AC, BM CM,
B C
You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are , so RUQ TUS by AAS.
RQ TS, UQ US, R T
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3. 1 and 2 are the base angles.
Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
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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4. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
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5. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”
Reading Math
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6. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
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7. A corollary is a statement that follows immediately from a theorem.
Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
A corollary is a statement that follows immediately from a theorem.
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8. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
Using the Isosceles Triangle Theorems
Explain why ABC is isosceles.
ABC and XAB are alternate interior angles formed by XA, BC, and the transversal AB. Because XA || BC, ABC XAB.
The diagram shows that XAB ACB. By the Transitive Property of Congruence, ABC ACB.
You can use the Converse of the Isosceles Triangle Theorem to conclude that AB AC.
By the definition of an isosceles triangle, ABC is isosceles.
Quick Check
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9. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
Using Algebra
Suppose that mL = y. Find the values of x and y.
MO LN The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
x = 90 Definition of perpendicular
mN = mL Isosceles Triangle Theorem
mL = y Given
mN = y Transitive Property of Equality
mN + mNMO + mMON = 180 Triangle Angle-Sum Theorem
y + y + 90 = 180 Substitute.
2y + 90 = 180 Simplify.
2y = 90 Subtract 90 from each side.
y = 45 Divide each side by 2.
Quick Check
Therefore, x = 90 and y = 45.
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10. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
Real-World Connection
Suppose the raised garden bed is a regular hexagon. Suppose that a segment is drawn between the endpoints of the angle marked x. Find the angle measures of the triangle that is formed.
Because the garden is a regular hexagon, the sides have equal length, so the triangle is isosceles.
By the Isosceles Triangle Theorem, the unknown angles are congruent.
Example 4 found that the measure of the angle marked x is 120. The sum of the angle measures of a triangle is 180.
If you label each unknown angle y, y + y = 180.
y = 180
2y = 60
y = 30
Quick Check
So the angle measures in the triangle are 120, 30 and 30.
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11. Isosceles and Equilateral Triangles
GEOMETRY LESSON 4-5
Use the diagram for Exercises 1–3.
1. If mBAC = 38, find mC.
2. If mBAM = mCAM = 23, find mBMA.
3. If mB = 3x and mBAC = 2x – 20, find x.
4. Find the values of x and y. 71 90 25
5. ABCDEF is a regular hexagon. Find mBAC.
x = 60
y = 9 30
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