1. Slide 9 - 1 Copyright © 2009 Pearson Education, Inc. Unit 6 MM-150 SURVEY OF MATHEMATICS – Jody Harris Geometry

2. Slide 9 - 2 Copyright © 2009 Pearson Education, Inc. Basic Terms

3. Slide 9 - 3 Copyright © 2009 Pearson Education, Inc. Angles An angle is the union of two rays with a common endpoint; denoted The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle:

4. Slide 9 - 4 Copyright © 2009 Pearson Education, Inc. Angles The measure of an angle is the amount of rotation from its initial to its terminal side. Angles are classified by their degree measurement.  Right Angle is 90 o  Acute Angle is less than 90 o  Obtuse Angle is greater than 90 o but less than 180 o  Straight Angle is 180 o

5. Slide 9 - 5 Copyright © 2009 Pearson Education, Inc. Types of Angles ____ Straight Angle ____ Right Angle ____ Obtuse Angle ____ Acute Angle

6. Slide 9 - 6 Copyright © 2009 Pearson Education, Inc. Types of Angles Adjacent Angles - angles that have a common vertex and a common side but no common interior points. Complementary Angles - two angles whose sum of their measures is 90 degrees. Supplementary Angles - two angles whose sum of their measures is 180 degrees.

7. Slide 9 - 7 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D  ABC and  CBD

8. Slide 9 - 8 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D  ABC and  CBD  ABC +  CBD = 180

9. Slide 9 - 9 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D  ABC and  CBD  ABC +  CBD = 180 x

10. Slide 9 - 10 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D  ABC and  CBD  ABC +  CBD = 180 x 4x

11. Slide 9 - 11 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D x 4x  ABC and  CBD  ABC +  CBD = 180

12. Slide 9 - 12 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D x 4x  ABC and  CBD 4x + x = 180  ABC +  CBD = 180

13. Slide 9 - 13 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D x 4x  ABC and  CBD 4x + x = 180 5x = 180  ABC +  CBD = 180

14. Slide 9 - 14 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D x 4x  ABC and  CBD 4x + x = 180 5x = 180 x = 36  ABC +  CBD = 180

15. Slide 9 - 15 Copyright © 2009 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle. A B C D x 4x  ABC and  CBD 4x + x = 180 5x = 180 x = 36  ABC +  CBD = 180  ABC = 4x = 4(36) = 144

16. Slide 9 - 16 Copyright © 2009 Pearson Education, Inc. Example Find x. Assume that angle 1 and angle 2 are complementary angles.

17. Slide 9 - 17 Copyright © 2009 Pearson Education, Inc. Example Find x. Assume that angle 1 and angle 2 are complementary angles. x + 5x + 3 = 90

18. Slide 9 - 18 Copyright © 2009 Pearson Education, Inc. Example Find x. Assume that angle 1 and angle 2 are complementary angles. x + 5x + 3 = 90 6x + 3 = 90

19. Slide 9 - 19 Copyright © 2009 Pearson Education, Inc. Example Find x. Assume that angle 1 and angle 2 are complementary angles. x + 5x + 3 = 90 6x + 3 = 90 6x = 87

20. Slide 9 - 20 Copyright © 2009 Pearson Education, Inc. Example Find x. Assume that angle 1 and angle 2 are complementary angles. x + 5x + 3 = 90 6x + 3 = 90 6x = 87 x = 14.5

21. Slide 9 - 21 Copyright © 2009 Pearson Education, Inc. More definitions Vertical angles are the nonadjacent angles formed by two intersecting straight lines. Vertical angles have the same measure. A line that intersects two different lines, at two different points is called a transversal. Special angles are given to the angles formed by a transversal crossing two parallel lines.

22. Slide 9 - 22 Copyright © 2009 Pearson Education, Inc. Special Names 5 6 12 4 87 3 One interior and one exterior angle on the same side of the transversal–have the same measure Corresponding angles Exterior angles on the opposite sides of the transversal–have the same measure Alternate exterior angles Interior angles on the opposite side of the transversal–have the same measure Alternate interior angles 5 6 12 4 87 3 5 6 12 4 87 3

23. Slide 9 - 23 Copyright © 2009 Pearson Education, Inc. Types of Triangles Acute Triangle All angles are acute. Obtuse Triangle One angle is obtuse.

24. Slide 9 - 24 Copyright © 2009 Pearson Education, Inc. Types of Triangles continued Right Triangle One angle is a right angle. Isosceles Triangle Two equal sides. Two equal angles.

25. Slide 9 - 25 Copyright © 2009 Pearson Education, Inc. Types of Triangles continued Equilateral Triangle Three equal sides. Three equal angles (60º) each. Scalene Triangle No two sides are equal in length.

26. Slide 9 - 26 Copyright © 2009 Pearson Education, Inc. Similar Figures Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. 4 3 4 6 66 9 4.5

27. Slide 9 - 27 Copyright © 2009 Pearson Education, Inc. Similar Figures Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. 4 3 4 6 66 9 4.5 Put brown inside green

28. Slide 9 - 28 Copyright © 2009 Pearson Education, Inc. Example Catherine Johnson wants to measure the height of a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse? x 75 12 5

29. Slide 9 - 29 Copyright © 2009 Pearson Education, Inc. Quadrilaterals Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360 o. Quadrilaterals may be classified according to their characteristics.

30. Slide 9 - 30 Copyright © 2009 Pearson Education, Inc. Classifications Trapezoid Two sides are parallel. Parallelogram Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

31. Slide 9 - 31 Copyright © 2009 Pearson Education, Inc. Classifications continued Rhombus Both pairs of opposite sides are parallel. The four sides are equal in length. Rectangle Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.

32. Slide 9 - 32 Copyright © 2009 Pearson Education, Inc. Classifications continued Square Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

33. Slide 9 - 33 Copyright © 2009 Pearson Education, Inc. Formulas P = s 1 + s 2 + b 1 + b 2 P = s 1 + s 2 + s 3 P = 2b + 2w P = 4s P = 2l + 2w Perimeter Trapezoid Triangle A = bhParallelogram A = s 2 Square A = lwRectangle AreaFigure