1. Angle Relationships in Triangles
Geometry H2 (Holt 4-3) K. Santos

2. Triangle Sum Theorem (4-3-1)
The sum of the angle measures of a triangle is 180°. m< A + m<B + m<C =180°. B A C

3. Corollary
A corollary is a theorem whose proof follows directly from another theorem. “Mini-theorem” For example—there are two corollaries to the Triangle Sum Theorem

4. Corollary 1 (4-3-2)
The acute angles of a right triangle are complementary. D E F m< D and m< E are complementary m< D + m< E = 90°

5. Example
One of the acute angles in a right triangle measures 27°. What is the measure of the other acute angle?

6. Example
Find x, y and z yx 54° z 62°

7. Corollary 2 (4-3-3)
The measure of each angle of an equiangular triangle is 60°. A B C

8. Interior vs Exterior
2 Exterior Interior Exterior angle Interior angles Remote interior angles to <4

9. Exterior Angle Theorem (4-3-4)
The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles Exterior angle = sum of remote interior angles

10. Example 1 (exterior angle)
Find x. 53° x

11. Example 2 (remote interior angle)
Find x. x

12. Example 3 (algebraic)
H Find m< J and m< H. 4x ° 5x + 2 F G J

13. Third Angles Theorem (4-3-5)
If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. L R N M S T If: < L ≅ <R < M ≅ < S Then: < N ≅ <T

14. Example—Third angles
F Find m< C and m< F. A m< C = 4y - 8 m< F = 2y +10 B C D E