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Recall from 5-1 that an isosceles triangle has at least two congruent sides.

There are two theorems dealing with isosceles triangles

THEOREM 6-2: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

THEOREM 6-3: The median from the vertex angle of an isosceles triangle IS ALSO the perpendicular bisector of the base AND IS ALSO the angle bisector of the vertex angle.

Example Find the value of each variable in isosceles

triangle DEF if EG is an angle bisector.▪ Ignore the bisector for a

second…▪ This is an isosceles triangle▪ Angles opposite equal sides are

equal▪ x = 49

▪ Bring the bisector back in▪ The angle bisector of an isosceles triangle is also the

perpendicular bisector▪ y = 90

YOUR TURN Find the value of the variables in each

triangle

x = 65˚y = 50˚

x = 90˚y = 70˚

THEOREM 6-4: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Example In ABC, A B and mA = 48. Find mC, AC, and BC.▪ Finding mC▪ A B, so B also equals 48.▪ 180˚ in a triangle.▪ 48 + 48 + C = 180▪ 96 + C = 180▪ C = 84

Example In ABC, A B and mA = 48. Find mC, AC, and BC.▪ Finding AC & BC▪ This is an isosceles triangle, so

the two marked sides are equal.▪ 4x = 6x – 5▪ -2x = -5

▪ x = 5/2 (or 2.5)

▪ Plug back in to get AC/BC▪ AC = 4(2.5) = 10 BC = 6(2.5) – 5 = 10

THEOREM 6-5: A triangle is only equilateral if it is equiangular

Assignment Study Guide #6-4 and

Practice Masters #6-4