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Proving the Vertical Angles Theorem

Proving the Vertical Angles Theorem. Adapted from Walch Education. Key Concepts. Angles can be labeled with one point at the vertex, three points with the vertex point in the middle, or with numbers.

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Proving the Vertical Angles Theorem

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  1. Proving the Vertical Angles Theorem Adapted from Walch Education

  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. 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

  3. Straight Angle • Straight angles are straight lines. 1.8.1: Proving the Vertical Angles Theorem

  4. Adjacent Angles • 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

  5. 1.8.1: Proving the Vertical Angles Theorem

  6. 1.8.1: Proving the Vertical Angles Theorem

  7. 1.8.1: Proving the Vertical Angles Theorem

  8. Linear pair • Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angle 1.8.1: Proving the Vertical Angles Theorem

  9. 1.8.1: Proving the Vertical Angles Theorem

  10. Vertical Angles • Vertical angles are nonadjacent angles formed by two pairs of opposite rays. 1.8.1: Proving the Vertical Angles Theorem

  11. 1.8.1: Proving the Vertical Angles Theorem

  12. 1.8.1: Proving the Vertical Angles Theorem

  13. Angle Addition Postulate 1.8.1: Proving the Vertical Angles Theorem

  14. Key Concepts • 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

  15. Supplementary Angles • In the diagram below, the angles form a linear pair. • m∠ABD+m∠DBC= 180 1.8.1: Proving the Vertical Angles Theorem

  16. Supplementary Angles • Apair of supplementary angles that are nonadjacent. • m∠PQR+m∠TUV= 180 1.8.1: Proving the Vertical Angles Theorem

  17. 1.8.1: Proving the Vertical Angles Theorem

  18. Properties of Congruence 1.8.1: Proving the Vertical Angles Theorem

  19. Perpendicular Lines • Perpendicular lines form four adjacent and congruent right angles. 1.8.1: Proving the Vertical Angles Theorem

  20. Perpendicular Lines • The symbol for writing perpendicular lines is , and is read as “is perpendicular to.” • 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

  21. Concepts, continued • 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

  22. 1.8.1: Proving the Vertical Angles Theorem

  23. 1.8.1: Proving the Vertical Angles Theorem

  24. Complementary Angles • Complementary angles are two angles whose sum is 90º. • Complementary angles can form a right angle or be nonadjacent. 1.8.1: Proving the Vertical Angles Theorem

  25. Complementary Angles 1.8.1: Proving the Vertical Angles Theorem

  26. Complementary Angles • The diagram at right shows a pair of adjacent complementary angles labeled with numbers. 1.8.1: Proving the Vertical Angles Theorem

  27. 1.8.1: Proving the Vertical Angles Theorem

  28. Practice • Prove that vertical angles are congruent given a pair of intersecting lines, and . 1.8.1: Proving the Vertical Angles Theorem

  29. Draw a Diagram 1.8.1: Proving the Vertical Angles Theorem

  30. Supplement Theorem • Supplementary angles add up to 180º. 1.8.1: Proving the Vertical Angles Theorem

  31. 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

  32. Reflexive Property • m∠2 = m∠2 1.8.1: Proving the Vertical Angles Theorem

  33. 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

  34. 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

  35. Try this one… • 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

  36. Thanks For Watching! Ms. Dambreville

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