1. 4-6 Isosceles And Equilateral Triangles
You identified isosceles and equilateral triangles.
Use properties of isosceles triangles.
Use properties of equilateral triangles.

2. Isosceles Triangles Parts
vertex
Vertex angle
leg
leg
Base angles
base

3. 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?

4. Isosceles Triangle Theorem
If 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.

5. Page 285

6. A. Name two unmarked congruent angles.
BCA is opposite BA and A is opposite BC, so BCA  A. __
____
Answer: BCA and A

7. B. Name two unmarked congruent segments.
Answer: BC  BD

8. Page 286

9. Page 286

10. A. Find mR.
Since QP = QR, QP  QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR.
Triangle Sum Theorem
mQ = 60, mP = mR
Simplify.
Subtract 60 from each side.
Answer: mR = 60
Divide each side by 2.

11. B. Find PR.
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.
Answer: PR = 5 cm

12. A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°

13. 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°.

14. mDFE = 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

15. Try It B
What else must be true? A C O
3x+8
4x−10 M N

16. 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.

17. 4-6 Assignment
Page 289, 1-2, 15-22, 29-32