Dan opowerpoint


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Triangles and Angles

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Dan opowerpoint

  1. 1. Section 1.5<br />Triangles & Angles<br />10th Grade Geometry<br />Mr. O’Neill’s Class<br />
  2. 2. Standards & Objectives<br />Standard :<br />Students will learn and apply geometric concepts.<br />Objectives:<br />Classify triangles by their sides and angles.<br />Find angle measures in triangles<br />DEFINITION:<br /> A triangle is a figure formed by three segments joining three non-collinear points. <br />
  3. 3. Names of Triangles<br />Triangles can be classified by the sides or by the angle<br />Scalene—no congruent sides<br />Equilateral—3 congruent sides<br />Isosceles—2 congruent sides<br />
  4. 4. Equilateral Triangle<br />3 congruent angles, an equilateral triangle is also acute<br />
  5. 5. Acute Triangle<br />3 Acute Angles<br />
  6. 6. Parts of a Triangle<br />Each of the three points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices.<br />Two sides sharing a common vertex are adjacent sides. <br />The third is the side opposite an angle<br />adjacent<br />opposite<br />adjacent<br />
  7. 7. <ul><li>Red represents the hypotenuse of a right triangle. The sides that form the right angle are the legs.</li></ul>Right Triangles<br />leg<br />Hypotenuse<br />leg<br />
  8. 8. Finding Angle Measures<br />Corollary to the triangle sum theorem<br />The acute angles of a right triangle are complementary.<br />m A + m B =90<br />2X<br />X<br />
  9. 9. Finding Angle Measures<br />B<br />2X<br />X + 2x = 90<br />3x = 90<br />X = 30<br />So m A = 30 and the m B=60<br />X<br />A<br />C<br />
  10. 10. Angle Bisector<br /><ul><li>A ray that divides an angle into 2 congruent adjacent angles.</li></ul> BD is an angle bisector of <ABC.<br />A<br />B<br />D<br />C<br />
  11. 11. Reason:<br />Given<br />Def. Cong. Angles<br />Def. Cong. Angles<br />Transitive property<br />Def. Cong. Angles<br />Statement:<br />A ≅ B, B ≅ C<br />mA= mB<br />mB= mC<br />mA= mC<br />A ≅ C<br />Ex. 1: Transitive Property of Angle Congruence<br />
  12. 12. Using the Transitive Property<br />Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3<br />Prove: m1 =40<br />
  13. 13. All right angles are congruent.<br />Example 3: Proving Theorem 2.3<br />Given: 1 and 2 are right angles<br />Prove: 1 ≅ 2<br />Theorem 2.0<br />
  14. 14. Theorem 2.1: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.<br />If m1 +m2 = 180 AND m2 +m3 = 180, then 1 ≅ 3.<br />Properties of Special Pairs of Angles<br />
  15. 15. Theorem 2.2: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent.<br />If m4 +m5 = 90 AND m5 +m6 = 90, then 4 ≅ 6.<br />Congruent Complements Theorem<br />
  16. 16. Median of a Triangle<br />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<br />
  17. 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. <br />Center of the Triangle<br />
  18. 18. Comparing Measurements of a Triangle<br /><ul><li>After practicing a few exercises, you may have discovered a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles.</li></li></ul><li>Opposite Theorem<br />If one side of a triangle is longer than another side, then the angle oppositethe longer side is larger than the angleoppositethe shorter side.<br />mA >mC<br />
  19. 19. Distance Example<br />AB + BC > AC<br />MC + CG > MG<br />99 + 165 > x<br />264 > x<br />x + 99 < 165<br />x < 66<br /> 66 < x < 264<br />http://www.wolframalpha.com/examples/Geometry.html<br />