1. 4.8 – Use Isosceles and Equilateral Triangles
Geometry Chapter 4
4.8 – Use Isosceles and Equilateral Triangles

2. Warm-up 120° Classify the triangles by their sides AND angles. 1.) 2.)
1.) )
3.) )
120°

3. Apply Triangle Sum Properties
Objective: Students will be able to apply postulates and theorems of isosceles and equilateral triangles to find measures.
Agenda
Isosceles Triangles
Isosceles Triangle Theorems/Corollaries
Practice

4. Isosceles Triangles An Isosceles Triangle has the following properties
2 Congruent Sides (known as the legs)
1 Side with its own measure (known as the base)
The angle included between the legs is known as the vertex angle
Angles connected to the base are known as the base angles
Vertex Angle
Leg
Leg
Base Angles
Base

5. Isosceles Triangle Theorems
Theorem 4.7 – Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent 𝑨 𝑩 𝑪
If 𝐴𝐵 ≅ 𝐴𝐶
Then <𝐵≅ <𝐶

6. Isosceles Triangle Theorems
Theorem 4.8 – Converse of Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent 𝑨 𝑩 𝑪
If <𝐵≅ <𝐶
Then 𝐴𝐵 ≅ 𝐴𝐶

7. Example 1
a.) In ∆𝐷𝐸𝐹, 𝐷𝐸 ≅ 𝐷𝐹 . Name two congruent angles. b.) Using the same triangle ∆𝐷𝐸𝐹, find 𝑚<𝐸 and 𝑚<𝐹 if 𝑚<𝐷=30°. 𝑫 𝑬 𝑭

8. Example 1
a.) In ∆𝐷𝐸𝐹, 𝐷𝐸 ≅ 𝐷𝐹 . Name two congruent angles. <𝑬≅ <𝑭 𝑫 𝑬 𝑭

9. Example 1
b.) Using the same triangle ∆𝐷𝐸𝐹, find 𝑚<𝐸 and 𝑚<𝐹 if 𝑚<𝐷=30° If 𝑥=𝑚<𝐸=𝑚<𝐹, then 𝑥+𝑥+30=180 2𝑥+30=180 2𝑥=150 𝒙=𝟕𝟓° 𝑫 𝑬 𝑭

10. Example 1+
a.) In ∆𝑃𝑄𝑅, 𝑃𝑄 ≅ 𝑄𝑅 . Name two congruent angles. b.) Using the same triangle ∆𝐷𝐸𝐹, find 𝑚<𝑃 and 𝑚<𝑅 if 𝑚<𝑄=45°. 𝑹 𝑸 𝑷

11. Example 2
Use the given triangle to complete the statements a.) If 𝐻𝐺 ≅ 𝐻𝐾 , then < ________ ≅ < ________. b.) If <𝐾𝐻𝐽≅ <𝐾𝐽𝐻, then ________ ≅ ________. 𝑲 𝑮 𝑯 𝑱

12. Example 2
Use the given triangle to complete the statements a.) If 𝐻𝐺 ≅ 𝐻𝐾 , then <𝐆≅ <𝑮𝑲𝑯 b.) If <𝐾𝐻𝐽≅ <𝐾𝐽𝐻, then 𝑲𝑱 ≅ 𝑯𝑲 𝑲 𝑮 𝑯 𝑱

13. Example 3 Find the value of x in each of the following a.) b.) 𝒙° 𝟕𝟐°
𝟕𝟓°
𝟗𝒙°
𝟕𝟐°

14. Example 3 Solution: 75+75+𝑥=180 150+𝑥=180 𝒙=𝟑𝟎 Solution: 9𝑥=72 𝒙=𝟖
Find the value of x in each of the following a.) b.) 𝒙°
𝟕𝟓°
𝟗𝒙°
𝟕𝟐°
Solution:
75+75+𝑥=180
150+𝑥=180
𝒙=𝟑𝟎
Solution:
9𝑥=72
𝒙=𝟖

15. Corollaries
The following two statements are corollaries to theorems 4.7 and 4.8:
Corollary to the Base Angles Theorem: An equilateral triangle is also equiangular.
Corollary to the Converse of Base Angles Theorem: An equiangular triangle is also equilateral, with three 60° angles 𝑨 𝑩 𝑪

16. Example 4
Find the values of x and y in the diagram. 𝑲 𝑵 𝑳 𝑴 𝒚 𝟒
𝒙+𝟏

17. Example 4 For y: 𝒚=𝟒 (Why?) Find the values of x and y in the diagram.𝑲 𝑵 𝑳 𝑴 𝒚 𝟒
𝒙+𝟏
For y:
𝒚=𝟒 (Why?)

18. Example 4 For x: 𝑥+1=4 (Why?) 𝒙=𝟑
Find the values of x and y in the diagram. 𝑲 𝑵 𝑳 𝑴 𝒚 𝟒
𝒙+𝟏
For x:
𝑥+1=4 (Why?)
𝒙=𝟑

19. Example 5
Find the values of x and y in the diagram.
𝒚+𝟒 𝒙 𝟕

20. Example 5 For x: 𝒙=𝟕 (Why?) Find the values of x and y in the diagram.
𝒚+𝟒 𝒙 𝟕

21. Example 5 For y: 𝑦+4=7 (Why?) 𝒚=𝟑
Find the values of x and y in the diagram.
For y:
𝑦+4=7 (Why?)
𝒚=𝟑
𝒚+𝟒 𝒙 𝟕

22. Example 6
Find the values of x and y in the diagram. 𝒙° 𝒚°

23. Example 6 𝒙° 𝒚° For x: 𝒙=𝟔𝟎° (Why?)
Find the values of x and y in the diagram.
For x:
𝒙=𝟔𝟎° (Why?) 𝒙° 𝒚°

24. Example 6 𝒙° 𝒚° 𝟑𝟎° For y: 30+30+𝑦=180 60+𝑦=180 𝒚=𝟏𝟐𝟎
Find the values of x and y in the diagram.
For y:
30+30+𝑦=180
60+𝑦=180
𝒚=𝟏𝟐𝟎 𝒙° 𝒚°
𝟑𝟎°

25. 𝑪 𝑩 𝑫 𝑨 𝑬

26. 𝟗𝒚° 𝒙° 𝒚°
(𝒙+𝟕)°
𝟓𝟓°