GEOMETRY LESSON 4-5GEOMETRY 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.

Isosceles and Equilateral TrianglesIsosceles and Equilateral Triangles

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Check Skills You’ll Need

CABC

AB

By the Triangle Exterior Angle Theorem, x = 75 + 30 = 105°.

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.

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

Corresponding parts of congruent triangles are congruent.

RQ TS, UQ US, R T

You are given two pairs of s, and AM AM by the Reflexive Prop., so ABM ACM by ASA.

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You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are , so RUQ TUS by AAS.

AB AC, BM CM, B C

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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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.

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”

Reading Math

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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A corollary is a statement that follows immediately from a theorem.

Explain why ABC is isosceles.

By the definition of an isosceles triangle, ABC is

isosceles.

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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.

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Quick Check

Using the Isosceles Triangle Theorems

Suppose that mL = y. Find the values of x and y.

mN =mLIsosceles Triangle Theorem

mL = y Given

mN + mNMO + mMON=180Triangle Angle-Sum Theorem

mN = yTransitive Property of Equality

y + y + 90 =180Substitute. 2y + 90 =180

Simplify. 2y = 90 Subtract 90 from each side.y = 45 Divide each side by 2.

Therefore, x = 90 and y = 45.

MO LNThe bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.x = 90Definition of perpendicular

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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Quick Check

Using Algebra

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, 120 + y + y = 180.120 + 2y = 180

2y = 60y = 30

So the angle measures in the triangle are 120, 30 and 30.

Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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.

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Quick Check

Real-World Connection

Use the diagram for Exercises 1–3.

1.If mBAC = 38, find mC.2.If mBAM = mCAM = 23, find mBMA. 3.If mB = 3x and mBAC = 2x – 20, find x.

4. Find the values of x and y. 5.ABCDEF is a regular hexagon. Find mBAC.

71

90

25

x = 60y = 9

30

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Isosceles and Equilateral TrianglesIsosceles and Equilateral TrianglesGEOMETRY LESSON 4-5GEOMETRY LESSON 4-5

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