1. 4.2 Angles of Triangles

2. Objectives Apply the Angle Sum Theorem
Apply the Exterior Angle Theorem

3. Theorem 4.1 – Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180°.
mX + mY + mZ = 180° X Y Z

4. Example 1: Find the missing angle measures.
Find first because the measure of two angles of the triangle are known.
Angle Sum Theorem
Simplify.
Subtract 117 from each side.

5. Example 1: Angle Sum Theorem Simplify. Subtract 142 from each side.
Answer:

6. Your Turn:
Find the missing angle measures.
Answer:

7. Theorem 4.2 – Third Angle Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangle are congruent.
Abbreviation: If 2 s of one Δ are  to 2 s of another Δ, then third s are .

8. Exterior Angles and Triangles
An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ).
The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3). 1 2 3 4

9. Theorem 4.3 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles m 1 = m 2 + m 3 1 2 3 4

10. Example 2: Find the measure of each numbered angle in the figure.
Exterior Angle Theorem
Simplify.
If 2 s form a linear pair, they are supplementary.
Substitution
Subtract 70 from each side.

11. Example 2: Exterior Angle Theorem Substitution
Subtract 64 from each side.
If 2 s form a linear pair, they are supplementary.
Substitution
Simplify.
Subtract 78 from each side.

12. Example 2: Angle Sum Theorem Substitution Simplify.
Subtract 143 from each side.
Answer:

13. Your Turn: Find the measure of each numbered angle in the figure.
Answer:

14. Corollaries
A corollary is a statement that can be easily proven using a theorem.
Corollary 4.1 – The acute s of a right ∆ are complementary.
Corollary 4.2 – There can be at most one right or obtuse  in a ∆.

15. Example 3:
GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.
Corollary 4.1
Substitution
Subtract 20 from each side.
Answer:

16. Your Turn:
The piece of quilt fabric is in the shape of a right triangle. Find if is 32.
Answer:

17. Assignment Geometry: Pg. 189 #11 - 35
Pre-AP Geometry: Pg. 189 #