1. 4.1 – Apply triangle sum properties
Geometry Chapter 4
4.1 – Apply triangle sum properties

2. Apply Triangle Sum Properties
Objective: Students will be able to classify triangles, and use those classifications to find the measures of their angles.
Agenda
Classifying Triangles (By Sides and Angles)
Triangle Sum
Exterior Angles

3. Triangle Properties
Each of the three points of a triangles is known as a vertex
From ∆𝐴𝐵𝐶, we can see that
Vertices: points A, B, and C.
Sides: 𝐴𝐵 , 𝐵𝐶 , 𝐶𝐴
Angles: <𝐴, <𝐵, <𝐶 A B C

4. Types of Triangles
A triangle can be classified by the number of congruent sides it has
Scalene Triangle
Equilateral Triangle
Isosceles Triangle
No Sides Congruent
At least Two Sides Congruent
All Sides Congruent

5. Types of Triangles
A triangle can also be classified by the angles present in them
Acute Triangle
Obtuse Triangle
3 Acute <‘s
1 obtuse <
Right Triangle
Equiangular Triangle
1 Right <
All Congruent <‘s

6. Types of Triangles
Example 1: Classify the following triangles based off their side lengths and/or angle measures. 𝟖 𝟓 𝟕 𝟓 𝟑
𝟗𝟐°
𝟑𝟕°
𝟕𝟕°
𝟔𝟒°

7. Angles of a Triangle – Interior Angles
When the sides of a triangle are extended, other angles are created.
The original angles of the triangle are called the interior angles.

8. The Sum of the Angles of a Triangle
Theorem 4.1 – Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180. A B C
𝑚<𝐴+𝑚<𝐵+𝑚<𝐶=180°

9. The Sum of the Angles of a Triangle
Example 2: Find the value of x.

10. The Sum of the Angles of a Triangle
Example 2: Find the value of x.
We can apply theorem 4.1:
57+𝑥+2𝑥=180
57+3𝑥=180
3𝑥=123
𝒙=𝟒𝟏

11. The Sum of the Angles of a Triangle
Find the value of x.
We can apply theorem 4.1:
𝑥+𝑥+10+87=180
97+2𝑥=180
2𝑥=83
𝒙=𝟒𝟏.𝟓
𝟖𝟕° 𝒙°
(𝒙+𝟏𝟎)°

12. Corollaries
A statement that can be proved easily by applying a theorem is often called a corollary of the theorem.
The following statement is a corollary of theorem 4-1 (The Triangle Sum Theorem):

13. Corollaries B A C 𝑚<𝐴+𝑚<𝐵=90°
A statement that can be proved easily by applying a theorem is often called a corollary of the theorem.
The following statement is a corollary of theorem 4-1 (The Triangle Sum Theorem):
Corollary: The acute angles of a right triangle are complementary. B A C
𝑚<𝐴+𝑚<𝐵=90°

14. Corollaries
Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩. B A C 𝑥°
2𝑥°

15. Corollaries
Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩.
We can use the corollary:
𝑥+2𝑥=90
3𝑥=90
𝒙=𝟑𝟎° B A C 𝑥°
2𝑥°

16. Corollaries
Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩.
For 𝒎<𝑨:
𝑚<𝐴=𝑥=30° B A C 𝑥°
2𝑥°
For 𝒎<𝑩:
𝑚<𝐵=2𝑥=60°

17. Corollaries Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩.
We can use the corollary:
𝑥+𝑥=90
2𝑥=90
𝒙=𝟒𝟓° B A C 𝑥°
For 𝒎<𝑨:
𝑚<𝐴=𝑥=45°
For 𝒎<𝑩:
𝑚<𝐵=𝑥=45°

18. Angles of a Triangle – Interior Angles
When the sides of a triangle are extended, other angles are created.
The angles created by the extended lines, outside the triangle, are called exterior angles.
These angles form a linear pair with an interior angle of the triangle.

19. Exterior Angles Continued
Theorem 4.2 – Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles. 𝟏 A B C
𝑚<1=𝑚<𝐴+𝑚<𝐵

20. Exterior Angles
Example 4: Find the value of x. Then find the measure of the exterior angle.

21. Exterior Angles
Example 4: Find the value of x. Then find the measure of the exterior angle.
We can apply theorem 4.2:
5𝑥=120+𝑥
4𝑥=120
𝒙=𝟑𝟎

22. Exterior Angles
Example 4: Find the value of x. Then find the measure of the exterior angle.
For The Exterior Angle:
5𝑥=5(30)
=𝟏𝟓𝟎

23. Exterior Angles Find the value of x. Then find 𝑚<𝐽𝐾𝑀
We can apply theorem 4.2:
𝑥+70=2𝑥−5
𝒙=𝟕𝟓
𝟕𝟎° 𝑲 𝑱 𝑳 𝒙°
(𝟐𝒙−𝟓)° 𝑴
For 𝒎<𝑱𝑲𝑴:
𝑚<𝐽𝐾𝑀=2 75 −5
=150−5=𝟏𝟒𝟓

24. 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥.
𝟖𝟓° 𝒙°
(𝟐𝒙−𝟏𝟓)°
1.)
2.)
𝟏𝟑𝟔° 𝒙°
𝟑𝒙°