4.1 Triangles and Angles

1. 4.1 Triangles and Angles

2. Objectives/Assignments • Classify triangles by their sides and angles • Find angle measures in triangles • Assignment: 2-26 even, 32-38 even

3. 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

4. Acute Triangle Classification by Angles 3 acute angles

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

6. 1 right angle 1 obtuse angle Right Triangle Obtuse Triangle

7. 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 Parts of a Triangle When you classify a triangle, you need to be as specific as possible. adjacent Side opposite A adjacent

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

9. 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. Isosceles Triangles leg base leg

10. 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. Identifying the Parts of an Isosceles Triangle About 7 ft. 5 ft 5 ft

11. 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. Identifying the parts of an isosceles triangle (cont.) Hypotenuse & Base About 7 ft. 5 ft 5 ft leg leg

12. 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.

13. 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

14. 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)

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

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