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Proving Angle Relationships: Supplementary, Complementary, and Congruent Angles

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This guide explores the essential concepts of angle relationships in geometry, focusing on supplementary, complementary, and congruent angles. Learn how to write formal proofs involving these angle types, using the Protractor Rule and the Angle Addition Postulate. The document includes practical examples and proofs that demonstrate key theorems in action, ensuring a solid understanding of how to establish relationships between angles, including the properties of vertical angles and the principles of linear pairs. Perfect for mastering angle proofs!

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Proving Angle Relationships: Supplementary, Complementary, and Congruent Angles

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  1. 2.8 Proving Angle Relationships

  2. Objectives • Write proofs involving supplementary and complementary angles • Write proofs involving congruent and right angles

  3. Postulates Postulates 2.10 (Protractor Rule) All angles have measures between 0° and 180°. Postulate 2.11 (Angle Addition Postulate) R is in the interior of PQS iffmPQR + mRQS = mPQS. P R Q S

  4. If Example 1 is a right angle, find Answer: 50

  5. are supplementary and If and find Example 2: Supplement Theorem Subtraction Property Answer: 14

  6. If are complementary angles and . and find Your Turn: Answer: 28

  7. If 1 and 2are vertical angles and m1 andm2 find m1 and m2. 1 2 Vertical Angles Theorem m1 m2 Definition of congruent angles Example 3: Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3.

  8. In the figure, form a linear pair, and Prove that are congruent. Use a 2 column proof. and and Given: form a linear pair. Prove: Example 4:

  9. Proof: Statements Reasons 1. 1. Given 2. 2. Linear Pairs are supplementary 3. 3. Definition of supplementary angles 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Definition of congruent angles Example 3:  1 &  4 linear pair;

  10. Your Turn: In the figure, NYR and RYA form a linear pair,AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXYare congruent.

  11. Proof: Statements Reasons 1. 1. Given linear pairs. 2. 2.Linear pairs are supplementary. 3. 3.Given 4. 4. Your Turn:

  12. Assignment • Geometry WB pgs. 38-40

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