4-6 Isosceles And Equilateral Triangles

You identified isosceles and equilateral triangles.

• Use properties of isosceles triangles.

• Use properties of equilateral triangles.

Isosceles TrianglesParts

leg leg

base

Base angles

vertex

Vertex angle

The Isosceles Have It!An isosceles triangle has been drawn

on a piece of paper and then cut out. (How do you draw an isosceles triangle on a piece of paper?)

If the triangle is folded in half, what can be said about the base angles?

What can be said about the sides?

Isosceles Triangle TheoremIf two side of a triangle are congruent, then the

angles opposite those sides are congruent.

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Page 285

A. Name two unmarked congruent angles.

Answer: BCA and A

BCA is opposite BA and A is opposite BC, so BCA A.

______

B. Name two unmarked congruent segments.

Answer: BC BD

Page 286

Page 286

Since QP = QR, QP QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR.

A. Find mR.

Triangle Sum Theorem

mQ = 60, mP = mR

Simplify.

Subtract 60 from each side.

Divide each side by 2.Answer: mR = 60

Since all three angles measure 60, the triangle is equiangular. Because an equiangular triangle is also equilateral, QP = QR = PR. Since QP = 5, PR = 5 by substitution.

B. Find PR.

Answer: PR = 5 cm

A. 30°

B. 45°

C. 60°

D. 65°

A. Find mT.

ALGEBRA Find the value of each variable.

Since E = F, DE FE by the Converse of the Isosceles Triangle Theorem. DF FE, so all of the sides of the triangle are congruent. The triangle is equilateral. Each angle of an equilateral triangle measures 60°.

mDFE = 60 Definition of equilateral triangle

4x – 8 = 60 Substitution

4x = 68 Add 8 to each side.

x = 17 Divide each side by 4.

The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal.

DF = FE Definition of equilateral triangle

6y + 3 = 8y – 5 Substitution

3 = 2y – 5 Subtract 6y from each side.

8 = 2y Add 5 to each side. 4 = y Divide each side by 2.

Answer: x = 17, y = 4

Try It.,triangleisoscelesIn BCABABC

What else must be true?

.andsidesoflengths

theFind.,In

NOMO

NMMNO 3x+8 4x−10

M N

O

A

B

C

What makes an isosceles unique?

An isosceles triangle has two congruent sides and two congruent base angles.

What is an auxiliary line?

Auxiliary line is a line (or part of a line) added to a figure.

4-6 AssignmentPage 289, 1-2, 15-

22, 29-32