1. Isosceles and Equilateral Triangles

2. Isosceles Triangles The congruent sides are the LEGS.
The third side is the BASE.
The two congruent sides form the VERTEX ANGLE.
The other two angles are the BASE ANGLES.

3. Theorems 
Theorem 4.3: Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

4. Theorems 
Theorem 4.4: CONVERSE of Isosceles Triangle Theorem – If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

5. Theorems 
Theorem 4.5: The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

6. Proof of the Isosceles Triangle Theorem
Statements
Reasons 1. 2. 3. 4. 5.

7. Corollaries
corollary – a statement that follows immediately from a theorem
Corollary to Theorem 4.3: If a triangle is equilateral, then the triangle is equiangular.
Corollary to Theorem 4.4: If a triangle is equiangular, then the triangle is equilateral.

8. Applications
Find the value of y.

9. Solve for x and y

10. Solve for x

11. Solve for x

12. Solve for x

13. Ex 3. Find the Side Length of an Equiangular Triangle
Find the length of each side of the equiangular triangle. 3x
2x+10
Solution
Because the triangle is equiangular, it is also equilateral.
So,
3x=2x+10
x=10
3(10)=30
Each side of the triangle is 30

14. Definition of congruent segments
EX 3 Use Isosceles and Equilateral Triangles
Find the values of x and y in the diagram.
STEP 1
STEP 2
Definition of congruent segments
LN = LM
4 = x + 1
Substitute 4 for LN and x + 1 for LM.
3 = x
Subtract 1 from each side.

15. Practice Find the value of y. y=50 y=9 16 1. 2. 3. y+4=16 -4 -4 y=12 9
y=12 9 y 50 y
Y+4 B
5. Find the values of x and y in the diagram. 4. 58
y° = 120°
x° = 60°
6x+4 C A