Isosceles and Equilateral Triangles

1. Section 4-6
Isosceles and Equilateral Triangles

2. Essential Questions
❖ How do you use properties of isosceles triangles?
❖ How do you use properties of equilateral triangles?

3. Vocabulary
1. Legs of an Isosceles Triangle:
2. Vertex Angle:
3. Base Angles:

4. Vocabulary
1. Legs of an Isosceles Triangle: The two congruent sides
of an isosceles triangle
2. Vertex Angle:
3. Base Angles:

5. Vocabulary
1. Legs of an Isosceles Triangle: The two congruent sides
of an isosceles triangle
2. Vertex Angle: The included angle between the legs of
an isosceles triangle
3. Base Angles:

6. Vocabulary
1. Legs of an Isosceles Triangle: The two congruent sides
of an isosceles triangle
2. Vertex Angle: The included angle between the legs of
an isosceles triangle
3. Base Angles: The angles formed between each leg and
the base of an isosceles triangle

7. Theorems and Corollaries
Theorem 4.10 - Isosceles Triangle Theorem:
Theorem 4.11 - Converse of Isosceles Triangle Theorem:
Corollary 4.3 - Equilateral Triangles:
Corollary 4.4 - Equilateral Triangles:

8. Theorems and Corollaries
Theorem 4.10 - Isosceles Triangle Theorem: If two sides
of a triangle are congruent, then the angles opposite
those sides are congruent
Theorem 4.11 - Converse of Isosceles Triangle Theorem:
Corollary 4.3 - Equilateral Triangles:
Corollary 4.4 - Equilateral Triangles:

9. Theorems and Corollaries
Theorem 4.10 - Isosceles Triangle Theorem: If two sides
of a triangle are congruent, then the angles opposite
those sides are congruent
Theorem 4.11 - Converse of Isosceles Triangle Theorem:
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
Corollary 4.3 - Equilateral Triangles:
Corollary 4.4 - Equilateral Triangles:

10. Theorems and Corollaries
Theorem 4.10 - Isosceles Triangle Theorem: If two sides
of a triangle are congruent, then the angles opposite
those sides are congruent
Theorem 4.11 - Converse of Isosceles Triangle Theorem:
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
Corollary 4.3 - Equilateral Triangles: A triangle is
equilateral IFF it is equiangular
Corollary 4.4 - Equilateral Triangles:

11. Theorems and Corollaries
Theorem 4.10 - Isosceles Triangle Theorem: If two sides
of a triangle are congruent, then the angles opposite
those sides are congruent
Theorem 4.11 - Converse of Isosceles Triangle Theorem:
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
Corollary 4.3 - Equilateral Triangles: A triangle is
equilateral IFF it is equiangular
Corollary 4.4 - Equilateral Triangles: Each angle of an
equilateral triangle measures 60°

12. Example 1
a. Name two unmarked congruent angles.
b. Name two unmarked congruent
segments

13. Example 1
a. Name two unmarked congruent angles.
b. Name two unmarked congruent
segments

14. Example 1
a. Name two unmarked congruent angles.
b. Name two unmarked congruent
segments

15. Example 2
Find each measure.
a.
b. PR

16. Example 2
Find each measure.
180 - 60
a.
b. PR

17. Example 2
Find each measure.
180 - 60 = 120
a.
b. PR

18. Example 2
Find each measure.
180 - 60 = 120 120 ÷ 2
a.
b. PR

19. Example 2
Find each measure.
180 - 60 = 120 120 ÷ 2 = 60
a.
b. PR

20. Example 2
Find each measure.
180 - 60 = 120 120 ÷ 2 = 60
= 60°
a.
b. PR

21. Example 2
Find each measure.
180 - 60 = 120 120 ÷ 2 = 60
= 60°
a.
b. PR
Since all three angles will be 60°, this is an
equilateral triangle, so PR = 5 cm.

22. Example 3
Find the value of each variable.

23. Example 3
Find the value of each variable.
6y + 3 = 8y − 5

24. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y

25. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5

26. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5

27. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y

28. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22

29. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4

30. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8

31. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8

32. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0

33. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?

34. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?
4x − 8 = 60

35. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?
4x − 8 = 60
+ 8 + 8

36. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?
4x − 8 = 60
+ 8 + 8
4x = 68

37. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?
4x − 8 = 60
+ 8 + 8
4x = 68
44

38. Example 3
Find the value of each variable.
6y + 3 = 8y − 5
− 6y − 6y
3 = 2y − 5
+ 5 + 5
8 = 2y
22
y = 4
4x − 8 = 4x − 8
− 4x − 4x+ 8 + 8
0 = 0
Now what?
4x − 8 = 60
+ 8 + 8
4x = 68
44
x = 17