Explain why ABC is isosceles.

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# Explain why ABC is isosceles. - PowerPoint PPT Presentation

 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 .

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Presentation Transcript

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 ABAC.

By the definition of an isosceles triangle, ABC is isosceles.

Isosceles and Equilateral Triangles

LESSON 4-5

Explain why ABC is isosceles.

Quick Check

MOLNThe bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

x = 90 Definition of perpendicular

Isosceles and Equilateral Triangles

LESSON 4-5

Suppose that mL = y. Find the values of x and y.

mN = mLIsosceles Triangle Theorem

mL = y Given

mN = yTransitive 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.

Isosceles and Equilateral Triangles

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.

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 = 60

y = 30

Quick Check

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