1. 4.1 Triangles and Angles

2. Goal 1: Classifying Triangles
A triangle is a figure formed by three segments joining three noncollinear points.
Triangles can be classified by the sides or by the angle
Equilateral
3 congruent sides
Isosceles Triangle
2 congruent sides
Scalene
0 congruent sides

3. Classification by Angles
Acute Triangle
3 acute angles

4. Equiangular Triangle
3 congruent angles. An equiangular triangle is also acute.

5. Right Triangle
Obtuse Triangle
1 right angle
1 obtuse angle

6. Parts of a Triangle
When you classify a triangle, you need to be as specific as possible.
Each of the three points joining the sides of a triangle is a vertex. (plural: vertices). A, B and C are vertices.
Two sides sharing a common vertex are adjacent sides.
The third is the side opposite an angle
adjacent
Side opposite A
adjacent

7. Right Triangle
Red represents the hypotenuse of a right triangle, the side opposite the right angle. The sides that form the right angle are the legs.
hypotenuse
leg
leg

8. Isosceles Triangles
An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is the base.
leg
base
leg

9. Identifying the Parts of an Isosceles Triangle
Explain why ∆ABC is an isosceles right triangle.
In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle.
Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle.
About 7 ft.
5 ft
5 ft

10. Identifying the parts of an isosceles triangle (cont.)
Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle?
Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse.
Because AC  BC, side AB is also the base.
Hypotenuse & Base
About 7 ft.
5 ft
5 ft
leg
leg

11. Goal 2: Using Angle Measures of Triangles
Smiley faces are interior angles and hearts represent the exterior angles
Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.

12. Theorems Theorem 4.1: Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°
Theorem 4.2: Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary
Corollary: A statement that can be proved easily using the theorem

13. Example 3: Finding an Angle Measure
Exterior Angle Theorem: m1 = m A +m B
x + 65 = (2x + 10)
65 = x +10
55 = x
65(B)
(2x+10) (1)
x (A)

14. Finding Angle Measures
Corollary to the triangle sum theorem
The acute angles of a right triangle are complementary.
m A + m B = 90
2x (B)
X (A)

15. Finding Angle Measures (cont.)
X + 2x = 90
3x = 90
X = 30
So m A = 30 and the m B=60 B 2x A x C