1. 5.4 Isosceles and Equilateral Triangles
Geometry
What conjectures can you make about congruent angles and sides?

2. Geometry 5.4 Isosceles, Equilateral Triangles
Topic/Objective
Use properties of isosceles triangles.
Use properties of equilateral triangles.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

3. Opposite Angles and Sides
EF is opposite D.
E is opposite side DF. D F
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

4. Geometry 5.4 Isosceles, Equilateral Triangles
Isosceles Triangles
Vertex Angle
Leg
Leg
Base
Angles
Base
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

5. Geometry 5.4 Isosceles, Equilateral Triangles
A construction.
Begin with an isosceles triangle, ABC.
Draw the angle bisector from the vertex angle.
The angle bisector intersects the base at M.
ACM  BCM. Why?
SAS
A  B. Why?
CPCTC C A B M
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

6. Theorem 5.6 Base Angles Theorem.
If two sides of a triangle are congruent, then the angles opposite them are congruent.
(Easy form) The base angles of an isosceles triangle are congruent.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

7. Geometry 5.4 Isosceles, Equilateral Triangles
Visually:
This:
Means this:
The base angles of an isosceles triangle are congruent.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

8. Geometry 5.4 Isosceles, Equilateral Triangles
Example Problem
Solve for x.
x + x + 52 = 180
2x + 52 = 180
2x = 128
x = 64
52° x° x°
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

9. Geometry 5.4 Isosceles, Equilateral Triangles
Example 2
Solve for x and y.
In an isosceles triangle, base angles are congruent.
So y is…
42°
Now use the triangle angle sum theorem:
x = 180
x + 84 = 180
x = 96°
42° x° y°
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

10. Geometry 5.4 Isosceles, Equilateral Triangles
Example 3. You try it.
50° x° y°
x = 65°
y = 32.5°
Find x and y.
50° y°
2y = 180
2y = 65
y = 32.5° y°
32.5°`
115° x°
65°
65° x°
2x + 50 = 180
2x = 130
x = 65
180 – 65 = 115
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

11. Geometry 5.4 Isosceles, Equilateral Triangles
Example 4
Solve for x.
(2x)°
(3x – 25)°
3x – 25 = 2x
x = 25
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

12. Geometry 5.4 Isosceles, Equilateral Triangles
Theorem 5.7
Converse of the Base Angles Theorem.
If two angles of a triangle are congruent, then the sides opposite them are congruent.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

13. Geometry 5.4 Isosceles, Equilateral Triangles
Example 5
Since base angles are equal,
opposite sides are equal.
4x + 52 = 2x + 68
2x + 52 = 68
2x = 16
x = 8
4x + 52
2x + 68
4(8) + 52 = 84
Solve for x, then find the length of the legs.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

14. Geometry 5.4 Isosceles, Equilateral Triangles
Example 6
You do it.
Find the length of each side.
5x = 3x + 16
2x = 16
x = 8 40
4x – 2 5x 30
3x + 16 40
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

15. Equilateral Triangles Corollaries to Base Angles Theorem
If a triangle is equilateral, then it is also equiangular.
If a triangle is equiangular, then it is also equilateral.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

16. Geometry 5.4 Isosceles, Equilateral Triangles
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

17. Geometry 5.4 Isosceles, Equilateral Triangles
Example 7
Solve for x.
All sides are congruent.
3x – 10 = x + 10
2x = 20
x = 10
3x – 10
x + 10 2x
2x = x + 10
x = 10
3x – 10 = 2x
x – 10 = 0
x = 10
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

18. Geometry 5.4 Isosceles, Equilateral Triangles
One last problem. Solve for x and y. x° y°
50°
Solution…
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

19. Geometry 5.4 Isosceles, Equilateral Triangles
Solution…
This triangle is equilateral. Each angle is?
40° x°
70°
80° y°
60°
70° ?
50°
50°
60°
60°
These angles form straight angle. The missing angle is?
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles

20. Geometry 5.4 Isosceles, Equilateral Triangles
Summarize what you have learned today
The base angles of an isosceles triangle are congruent.
Equilateral triangles are Equiangular.
April 25, 2018
Geometry 5.4 Isosceles, Equilateral Triangles