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Mastering Angle Relationships Through Theorems and Postulates

Explore angle relationships with Ohio Content Standards, including proving conjectures, using inductive reasoning, and testing properties of geometric objects. Dive into postulates and theorems like Protractor Postulate, Angle Addition Postulate, and more. Practice problems included.

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Mastering Angle Relationships Through Theorems and Postulates

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  1. Lesson 2-8 Proving Angle Relationships

  2. Ohio Content Standards:

  3. Ohio Content Standards: Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.

  4. Ohio Content Standards: Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

  5. Ohio Content Standards: Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof.

  6. Postulate 2.10Protractor Postulate

  7. Postulate 2.10Protractor Postulate Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.

  8. Postulate 2.11Angle Addition Postulate

  9. Postulate 2.11Angle Addition Postulate

  10. At 4 o’clock, the angle between the hour and minute hands of a clock is 120°. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands?

  11. Theorem 2.3Supplement Theorem

  12. Theorem 2.3Supplement Theorem If two angles form a linear pair, then they are supplementary angles.

  13. Theorem 2.4Complement Theorem

  14. Theorem 2.4Complement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.

  15. If 1 and 2 form a linear pair and m 2 = 166, find m 1.

  16. Theorem 2.5

  17. Theorem 2.5 Congruence of angles is reflexive, symmetric, and transitive.

  18. Reflexive Property

  19. Reflexive Property

  20. Symmetric Property

  21. Symmetric Property

  22. Transitive Property

  23. Transitive Property

  24. Theorem 2.6

  25. Theorem 2.6 Angles supplementary to the same angle or to congruent angles are congruent.

  26. Theorem 2.7

  27. Theorem 2.7 Angles complementary to the same angle or to congruent angles are congruent.

  28. In the figure, 1 and 4 form a linear pair, and m 3 + m 1 = 180. Prove that 3 and 4 are congruent. 1 4 2 3

  29. Theorem 2.8Vertical Angle Theorem

  30. Theorem 2.8Vertical Angle Theorem If two angles are vertical angles, then they are congruent.

  31. If 1 and 2 are vertical angles and m 1 = d – 32 and m 2 = 175 – 2d, find m 1 and m 2.

  32. Assignment:Pgs. 112 - 114 16-38 evens, 46

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