Triangles and Angles

1. Section 1.5 Triangles & Angles 10th Grade Geometry Mr. O’Neill’s Class

2. Standards & Objectives Standard : Students will learn and apply geometric concepts. Objectives: Classify triangles by their sides and angles. Find angle measures in triangles DEFINITION: A triangle is a figure formed by three segments joining three non-collinear points.

3. Names of Triangles Triangles can be classified by the sides or by the angle Scalene—no congruent sides Equilateral—3 congruent sides Isosceles—2 congruent sides

4. Equilateral Triangle 3 congruent angles, an equilateral triangle is also acute

5. Acute Triangle 3 Acute Angles

6. Parts of a Triangle 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 opposite adjacent

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

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

11. Reason: Given Def. Cong. Angles Def. Cong. Angles Transitive property Def. Cong. Angles Statement: A ≅ B, B ≅ C mA= mB mB= mC mA= mC A ≅ C Ex. 1: Transitive Property of Angle Congruence

12. Using the Transitive Property Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 Prove: m1 =40

13. All right angles are congruent. Example 3: Proving Theorem 2.3 Given: 1 and 2 are right angles Prove: 1 ≅ 2 Theorem 2.0

14. Theorem 2.1: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If m1 +m2 = 180 AND m2 +m3 = 180, then 1 ≅ 3. Properties of Special Pairs of Angles

15. Theorem 2.2: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent. If m4 +m5 = 90 AND m5 +m6 = 90, then 4 ≅ 6. Congruent Complements Theorem

16. Median of a Triangle A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. For instance in ∆ABC, shown at the right, D is the midpoint of side BC. So, AD is a median of the triangle

17. The three medians of a triangle are concurrent (they meet). The point of concurrency is called the center of the triangle. The center, labeled P in the diagrams in the next few slides are ALWAYS inside the triangle. Center of the Triangle

19. Distance Example AB + BC > AC MC + CG > MG 99 + 165 > x 264 > x x + 99 < 165 x < 66 66 < x < 264 http://www.wolframalpha.com/examples/Geometry.html