GEOMETRYHELP

Explain why ABC is isosceles.

By the definition of an isosceles triangle, 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.

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Isosceles and Equilateral TrianglesLESSON 4-5

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GEOMETRYHELP

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 TrianglesLESSON 4-5

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GEOMETRYHELP

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°.

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.

Isosceles and Equilateral TrianglesLESSON 4-5

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