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Understanding Vertical Angle Theorem and Opposite Rays in Geometry

This lesson explores the concept of opposite rays and vertical angles in geometry. Opposite rays are defined as two collinear rays sharing a common endpoint and extending in opposite directions. Vertical angles occur when two angles are formed by intersecting rays, creating pairs of opposite angles. Examples illustrate how to identify vertical angles and apply their properties, including solving for unknown angle measures. Practice problems at the end reinforce understanding of these fundamental geometric concepts.

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Understanding Vertical Angle Theorem and Opposite Rays in Geometry

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  1. 2.8 Vertical Angle Theorem

  2. Opposite Rays • Def: Opposite rays are two collinear rays that have a common endpoint and extend in opposite directions C A A B B but not C

  3. Vertical Angles • Def: Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays ∠1 and ∠3 are vertical angles ∠2 and ∠4 are vertical angles 1 2 4 3

  4. Example 1 • Given: m∠1=37˚ • Find: m∠2, m∠3, m∠4 1 2 4 3

  5. Example 2 • Given: ∠4≅∠6 • Prove: ∠5≅∠6 6 5 4

  6. Example 3 • Given: ∠V≅∠YRX, ∠Y≅∠TRV • Prove: ∠V≅∠Y T V R Y X

  7. Example 4 • Is this possible? (-10x)˚ (-8x-10)˚

  8. Assignment • p. 102 2-7, 12

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