1. Isosceles and Equilateral Triangles
Section 4.3

2. Objectives Use properties of isosceles triangles
Use properties of equilateral triangles

3. Key Vocabulary Legs of an Isosceles Triangle
Base of an Isosceles Triangle
Vertex Angle
Base Angles

4. Theorems 4.3 Base Angles Theorem 4.4 Converse of Base Angles Theorem
4.5 Equilateral Theorem
4.6 Equiangular Theorem

5. Definitions Review Isosceles Triangle Equilateral Triangle
At least 2 congruent sides
From Greek: Isos – means “equal,” and – sceles means “leg.” So, isosceles means equal legs.
Equilateral Triangle
3 congruent sides
From Latin: Equi – means “equal,” and – lateral means “side.” So, equilateral means equal sides.
An equilateral triangle is a special case of an isosceles triangle having not just two, but all three sides equal.

6. Properties of Isosceles Triangles
The  formed by the ≅ sides is called the vertex angle.
The two ≅ sides are called legs. The third side is called the base.
The two s formed by the base and the legs are called the base angles.
vertex
leg
leg
base

7. Definitions - Review is an isosceles triangle. Name each item(s):
Vertex Angle
Base AC
Legs
AB, CB
Base Angles
Side opposite C AB
Angle opposite BC

8. Base Angles Theorem
Theorem 4.3 If two sides of a triangle are congruent, then the angles opposite those sides are congruent ().
If AC ≅ AB, then B ≅ C. A B C

9. The Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If B ≅ C, then AC ≅ AB. A B C

10. Example 1 Find the measure of L.
52 ?
SOLUTION
Angle L is a base angle of an isosceles triangle. From the Base Angles Theorem, L and N have the same measure.
ANSWER
The measure of L is 52°. 10

11. Example 2 Find the value of x. SOLUTION
By the Converse of the Base Angles Theorem, the legs have the same length.
DE = DF
Converse of the Base Angles Theorem
x + 3 = 12
Substitute x + 3 for DE and 12 for DF.
x = 9
Subtract 3 from each side.
ANSWER
The value of x is 9. 11

12. Your Turn:
Find the value of y. 1.
ANSWER 50 2.
ANSWER 9 3.
ANSWER 12

13. Example 3a:
Name two congruent angles.
Answer:

14. Example 3b: Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,
Answer:

15. Your Turn: a. Name two congruent angles. Answer:
b. Name two congruent segments.
Answer:

16. More Practice

17. ∠1≅∠3

18. ∠11≅∠8

21. ∠G

22. Solve for x and y x = 72 y + 72 +72 = 180 y + 144 = 180 y = 180 - 144

23. Solve for x and y ∠x≅∠1 m∠x + m∠1 = 90 2x = 90 x = 45 x + y = 180

24. Solve for x

25. Solve for x

26. Solve for x
x = 63

27. Your Turn - Find the missing measures (not drawn to scale)1. 2.
44° ? ?
30° ? ?

28. Find the missing measures (not drawn to scale)1.
The two base angles are = to each other b/c they are opposite congruent sides
180 – 44 = 136°
136/2 = 68°
44°
68°
68° ? ?

29. Find the missing measures (not drawn to scale)2. ? ?
30°

30. Find the missing measures (not drawn to scale)
The other base angle must be 30° b/c its opposite from a congruent side
180 – (30+30) = 120 2. ?
120° ?
30°
30°

31. Properties of Equilateral ∆’s
Equilateral Triangle – a triangle with three congruent sides.

32. Equilateral Theorem
Theorem 4.5 If a triangle is equilateral, then it is equiangular. C B A

33. Equiangular Theorem
Theorem 4.6 If a triangle is equiangular, then it is equilateral. C B A
60˚

34. Equilateral and Equiangular Theorems
What these theorems mean.
In a Triangle;
If all 3 sides are equal, then all 3 angles measure 60˚.
If all 3 angles measure 60˚, then all 3 sides are equal.
An equilateral triangle is an equiangular triangle and vice versa.

35. Example 4 SOLUTION Sides of an equilateral ∆ are congruent.
Find the length of each side of the equiangular triangle.
SOLUTION
The angle marks show that ∆QRT is equiangular. So, ∆QRT is also equilateral.
3x = 2x + 10
Sides of an equilateral ∆ are congruent.
x = 10
Subtract 2x from each side.
Substitute 10 for x.
3(10) = 30
ANSWER
Each side of ∆QRT is 30. 35

36. Example 5a: EFG is equilateral, and bisects bisects Find and
Since the angle was bisected,
Each angle of an equilateral triangle measures 60°.

37. Example 5a: is an exterior angle of EGJ. Exterior Angle Theorem
Substitution
Add.
Answer:

38. Example 5b: EFG is equilateral, and bisects bisects Find
Linear pairs are supplementary.
Substitution
Subtract 75 from each side.
Answer: 105

39. Your Turn: ABC is an equilateral triangle. bisects a. Find x.
Answer: 30 b.
Answer: 90

40. Your Turn:
Solve for x and y
x = 60
y = 120

41. Joke Time What has four legs and one arm? A happy pit bull.
What's the difference between chopped beef and pea soup?
Everyone can chop beef, but not everyone can pea soup!
What do you get when you cross an elephant and a rhino?
el-if-i-no

42. Assignment
Sec 4.3, Pg : #1 – 25 odd, 29 – 39 odd