1. Triangles and Angles
Section 4.1 and 4.2

2. Triangle Classifications by Sides
Equilateral Triangle:
3 sides that are congruent
Isosceles Triangle:
2 sides that are congruent
Scalene Triangle:
No sides that are congruent

3. Triangle Classifications by Angle
Acute Triangle:
3 angles that are acute
Equiangular Triangle
3 angles that are congruent
Right Triangle:
1 angle that is right
Obtuse Triangle:
1 angle that is obtuse

4. Parts of a Triangle Vertex: each of the three
points joining the sides of a triangle
Adjacent Sides: two sides
of a triangle that share a common vertex
Legs:
a. the sides that form the right angle
b. the two sides of an isosceles
triangle that are equal
Hypotenuse:
the side opposite the right angle
Base: the third side of an isosceles
triangle that is not equal
Adjacent Sides
Opposite Side
Hypotenuse
Legs
Base

5. Angles of a Triangle
Interior Angles: the three angles inside of the triangle
Exterior Angles: the angles
adjacent to the interior angles

6. Theorem 4.1: Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180◦.
m A + m B + m C = 180◦
Let’s see how that works!
(Assume the horizontal lines
are parallel.) A C B

7. Example m A + m B + m C = 180◦ x + 3x + 5x = 180◦ 9x = 180◦ x = 20◦ A

8. Example
m A + m B + m C = 180◦ 2x + x x - 5= 180◦ 7x - 9 = 180◦ 7x = 189◦ x = 27◦ A 2x
4x - 5
x - 4 C B

9. Theorem 4.2: Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
m 1 = m A + m B
Let’s see how that works! A 1 B C

10. Example
m 1 = m A + m B 10x - 3◦ = 4x + x + 12◦ 10x - 3◦ = 5x + 12◦ -5x +3◦ -5x +3◦ 5x = 15 ◦ x = 3 ◦ 1 B A C 4x
x + 12◦
10x - 3◦

11. Example
m 1 = m A + m B 6x - 10◦ = 2x + 3x + 2◦ 6x - 10◦ = 5x + 2◦ -5x +10◦ -5x + 10◦ x = 12 ◦ 1 B A C 2x
3x + 2◦
6x - 10◦

12. Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
m A + m B = 90◦
Definition: Corollaries to theorems
are statements that can be easily
proved using the theorem. A C B

13. Example
m A + m B = 90◦ 8x + x + 9◦ = 90◦ 9x + 9◦ = 90◦ - 9◦ -9◦ 9x = 81◦ x = 9◦ A 8x
x + 9◦ C B

14. Congruence and Triangles
Congruent Figures: they’re the same shape---their corresponding angles and corresponding sides are congruent and have the same corresponding measures. F B A H E D C G

15. Theorem 4.3: Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If A = D and B = E, then C = F. D A B F C E

16. Theorem 4.4: Properties of Congruent Triangles
Reflexive Property of Congruent Triangles:
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles:
If ABC = DEF, then DEF = ABC.
Transitive Property of Congruent Triangles:
If ABC = DEF and DEF = JKL, then ABC = JKL.

17. What We Learned Today Triangle Classifications by Sides
Triangle Classifications by Angle
Parts of a Triangle
Angles of a Triangle
Triangle Sum Theorem
Exterior Angle Theorem
Corollary to the Triangle Sum Theorem
Congruence and Triangles
Third Angles Theorem
Properties of Congruent Triangles