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Understanding Geometry: Postulates, Theorems, and Angle Relationships

This geometry lesson plan focuses on reviewing essential postulates and theorems through a practice quiz. Students will explore the relationships between angles, determining complementary and supplementary angles. Key concepts include naming the postulates that justify given statements and proving relationships among angles. The session includes warm-up solutions to build confidence and engagement, followed by clean-up reminders to encourage responsibility. Aim for a thorough understanding of angle properties and improve skills in reasoning and justification in geometry.

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Understanding Geometry: Postulates, Theorems, and Angle Relationships

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Review Asg #17 • Practice Quiz • Lesson: • Homework / Questions • Clean-up

  3. Practice Quiz (~5 min) • For each of these statements, name the postulate or theorem that justifies the statement: • If x + 1 = 2x, then 1 = x. • If x⁄3 – 6 = 3, then x⁄3 = 9. • If A + B = 70° and A + B + C = 180°,then 70° + C = 180°. • If X-Y-Z, then XY + YZ = XZ. • Let d be the distance between A and B.

  4. Warm-up Solutions • For each of these statements, name the postulate or theorem that justifies the statement: • If x + 1 = 2x, then 1 = x. The subtraction property • If x⁄3 – 6 = 3, then x⁄3 = 9. The addition property • If A + B = 70° and A + B + C = 180°,then 70° + C = 180°. The substitution property • If X-Y-Z, then XY + YZ = XZ.The Betweenness of Points Theorem • Let d be the distance between A and B.The Ruler Postulate

  5. Fields of Vision

  6. Popular Mechanics1902

  7. Definitions • Two angles are complementary iff their measures add up to 90°. • Each of these angles is the other’s complement. • Two angles are supplementary iff their measures add up to 180°. • Each of these is the other’s supplement.

  8. New Theorems • Complements of the same angle are of equal size. • Supplements of the same angle are of equal size.

  9. Theorem 3 – Proof Given: 1 and 2 are complements of 3. Prove: 1 = 2 Statement Reason • 1 and 2 are complements of 3. • 1 + 3 = 90°2 + 3 = 90° • 1 + 3 = 2 + 3 • 1 = 2 • Given. • Definition of complementary. • Substitution. • Subtraction (– 3)

  10. German geometry

  11. German geometry

  12. Definitions • Opposite rays point in opposite directions and share the same endpoint. • In other words, they form a straight angle (180°). • Two angles form a linear pair iff they share a common side and their other sides are opposite rays. • Two angles are vertical angles iff the sides of one are opposite rays to the sides of the other.

  13. Theorems • Theorem 5 (“Linear Pair Theorem”): • The angles in a linear pair are supplementary. • Theorem 6 (“Vertical Angles Theorem”): • Vertical angles are congruent.

  14. Homework • From Chapter, Section:

  15. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

  16. Practice Quiz • When you are finished, put away your pencil and take out two pens/pencils of different color for corrections. • First color = careless mistakes / “I knew that!” • Second color = things not understood / “I should review this!” • Come to my desk to pick up a solutions sheet,and check your work. • When you’re done with the solutions sheet, bring it back to me.

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