1 / 45

Understanding Angle Relationships in Geometry

Explore angle relationships in the same plane, such as vertical angles, adjacent angles, linear pairs, and perpendicular lines. Learn about supplementary and complementary angles. Practice using the Angle Addition Postulate and the Vertical Angles Theorem.

uttara
Download Presentation

Understanding Angle Relationships in Geometry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Introduction Think about crossing a pair of chopsticks and the angles that are created when they are opened at various positions. How many angles are formed? What are the relationships among those angles? This lesson explores angle relationships. We will be examining the relationships of angles that lie in the same plane. A plane is a two-dimensional figure, meaning it is a flat surface, and it extends infinitely in all directions. Planes require at least three non-collinear points. Planes are named using those points or a capital script letter. Since they are flat, planes have no depth. 1.8.1: Proving the Vertical Angles Theorem

  2. Key Concepts Angles can be labeled with one point at the vertex, three points with the vertex point in the middle, or with numbers. See the examples that follow. 1.8.1: Proving the Vertical Angles Theorem

  3. Key Concepts, continued Be careful when using one vertex point to name the angle, as this can lead to confusion. If the vertex point serves as the vertex for more than one angle, three points or a number must be used to name the angle. 1.8.1: Proving the Vertical Angles Theorem

  4. Key Concepts, continued Straight angles are angles with rays in opposite directions—in other words, straight angles are straight lines. 1.8.1: Proving the Vertical Angles Theorem

  5. Key Concepts, continued Adjacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points. Nonadjacent angles have no common vertex or common side, or have shared interior points. 1.8.1: Proving the Vertical Angles Theorem

  6. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  7. Key Concepts, continued (continued) 1.8.1: Proving the Vertical Angles Theorem

  8. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  9. Key Concepts, continued Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angle. 1.8.1: Proving the Vertical Angles Theorem

  10. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  11. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  12. Key Concepts, continued Vertical angles are nonadjacent angles formed by two pairs of opposite rays. 1.8.1: Proving the Vertical Angles Theorem

  13. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  14. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  15. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  16. Key Concepts, continued Informally, the Angle Addition Postulate means that the measure of the larger angle is made up of the sum of the two smaller angles inside it. Supplementary angles are two angles whose sum is 180º. Supplementary angles can form a linear pair or be nonadjacent. 1.8.1: Proving the Vertical Angles Theorem

  17. Key Concepts, continued In the diagram below, the angles form a linear pair. m∠ABD+m∠DBC= 180 1.8.1: Proving the Vertical Angles Theorem

  18. Key Concepts, continued The next diagram shows a pair of supplementary angles that are nonadjacent. m∠PQR+m∠TUV= 180 1.8.1: Proving the Vertical Angles Theorem

  19. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  20. Key Concepts, continued Angles have the same congruence properties that segments do. 1.8.1: Proving the Vertical Angles Theorem

  21. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  22. Key Concepts, continued Perpendicular lines form four adjacent and congruent right angles. 1.8.1: Proving the Vertical Angles Theorem

  23. Key Concepts, continued The symbol for indicating perpendicular lines in a diagram is a box at one of the right angles, as shown below. 1.8.1: Proving the Vertical Angles Theorem

  24. Key Concepts, continued The symbol for writing perpendicular lines is , and is read as “is perpendicular to.” In the diagram, . Rays and segments can also be perpendicular. In a pair of perpendicular lines, rays, or segments, only one right angle box is needed to indicate perpendicular lines. 1.8.1: Proving the Vertical Angles Theorem

  25. Key Concepts, continued Remember that perpendicular bisectors are lines that intersect a segment at its midpoint at a right angle; they are perpendicular to the segment. Any point along the perpendicular bisector is equidistant from the endpoints of the segment that it bisects. 1.8.1: Proving the Vertical Angles Theorem

  26. Key Concepts, continued (continued) 1.8.1: Proving the Vertical Angles Theorem

  27. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  28. Key Concepts, continued Complementary angles are two angles whose sum is 90º. Complementary angles can form a right angle or be nonadjacent. The following diagram shows a pair of nonadjacent complementary angles. 1.8.1: Proving the Vertical Angles Theorem

  29. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  30. Key Concepts, continued The diagram at right shows a pair of adjacent complementary angles labeled with numbers. 1.8.1: Proving the Vertical Angles Theorem

  31. Key Concepts, continued 1.8.1: Proving the Vertical Angles Theorem

  32. Common Errors/Misconceptions not recognizing the theorem that is being used or that needs to be used setting expressions equal to each other rather than using the Complement or Supplement Theorems mislabeling angles with a single letter when that letter is the vertex for adjacent angles not recognizing adjacent and nonadjacent angles 1.8.1: Proving the Vertical Angles Theorem

  33. Guided Practice Example 4 Prove that vertical angles are congruent given a pair of intersecting lines, and . 1.8.1: Proving the Vertical Angles Theorem

  34. Guided Practice: Example 4, continued Draw a diagram and label three adjacent angles. 1.8.1: Proving the Vertical Angles Theorem

  35. Guided Practice: Example 4, continued Start with the Supplement Theorem. Supplementary angles add up to 180º. 1.8.1: Proving the Vertical Angles Theorem

  36. Guided Practice: Example 4, continued Use substitution. Both expressions are equal to 180, so they are equal to each other. Rewrite the first equation, substituting m∠2 + m∠3 in for 180. m∠1 + m∠2 = m∠2 + m∠3 1.8.1: Proving the Vertical Angles Theorem

  37. Guided Practice: Example 4, continued Use the Reflexive Property. m∠2 = m∠2 1.8.1: Proving the Vertical Angles Theorem

  38. Guided Practice: Example 4, continued Use the Subtraction Property. Since m∠2 = m∠2, these measures can be subtracted out of the equation m∠1 + m∠2 = m∠2 + m∠3. This leaves m∠1 = m∠3. 1.8.1: Proving the Vertical Angles Theorem

  39. Guided Practice: Example 4, continued Use the definition of congruence. Since m∠1 = m∠3, by the definition of congruence, ∠1 and ∠3 are vertical angles and they are congruent. This proof also shows that angles supplementary to the same angle are congruent. ✔ 1.8.1: Proving the Vertical Angles Theorem

  40. Guided Practice: Example 4, continued 1.8.1: Proving the Vertical Angles Theorem

  41. Guided Practice Example 5 In the diagram at right, is the perpendicular bisector of . If AD = 4x – 1 and DC = x + 11, what are the values of AD and DC? 1.8.1: Proving the Vertical Angles Theorem

  42. Guided Practice: Example 5, continued Use the Perpendicular Bisector Theorem to determine the values of AD and DC. If a point is on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment being bisected. That means AD = DC. 1.8.1: Proving the Vertical Angles Theorem

  43. Guided Practice: Example 5, continued Use substitution to solve for x. 1.8.1: Proving the Vertical Angles Theorem

  44. Guided Practice: Example 5, continued Substitute the value of x into the given equations to determine the values of AD and DC. AD and DC are each 15 units long. ✔ 1.8.1: Proving the Vertical Angles Theorem

  45. Guided Practice: Example 5, continued 1.8.1: Proving the Vertical Angles Theorem

More Related