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Lesson 4.4 Angle Properties pp. 135-141

This lesson explores the properties of angles, focusing on linear pairs, vertical angles, complementary angles (summing to 90°), and supplementary angles (summing to 180°). Theorems such as the Vertical Angle Theorem and Linear Pair Theorem are proven, establishing relationships between these angles. Practice problems include finding complements, supplements, and measures of angles that form vertical and linear pairs. By the end of the lesson, students will have a stronger grasp of angle relationships and proofs in geometry.

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Lesson 4.4 Angle Properties pp. 135-141

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  1. Lesson 4.4 Angle Properties pp. 135-141

  2. Objectives: 1. To identify linear pairs and vertical, complementary, and supplementary angles. 2. To prove theorems on related angles.

  3. D A B C Definition A linear pair is a pair of adjacent angles whose noncommon sides form a straight angle (are opposite rays).

  4. Definition Vertical angles are angles adjacent to the same angle and forming linear pairs with it. E A B C D

  5. Definition Two angles are complementary if the sum of their measures is 90°. Two angles are supplementary if the sum of their measures is 180°.

  6. C Y 67° 23° T F X CFY and YFX are complementary

  7. C Y 157° 23° T F X TFY and YFX are supplementary

  8. Theorem 4.1 All right angles are congruent.

  9. STATEMENTSREASONS A and B are Given right angles 12. mA = 90° 12. _______________ mB = 90° 13. mA = mB 13. _______________ 14. A  B 14. _______________ Def. of rt. angle Substitution Def. of  angles

  10. Theorem 4.2 If two angles are adjacent and supplementary, then they form a linear pair.

  11. Theorem 4.3 Angles that form a linear pair are supplementary.

  12. Theorem 4.4 If one angle of a linear pair is a right angle, then the other angle is also a right angle.

  13. Theorem 4.5 Vertical Angle Theorem. Vertical angles are congruent.

  14. Theorem 4.6 Congruent supplementary angles are right angles.

  15. Theorem 4.7 Angle Bisector Theorem. If AB bisects CAD, then mCAB = ½mCAD.

  16. Practice: If the mA = 58°, find the measure of the supplement of A.

  17. Practice: If the mA = 58°, find the measure of the complement of A.

  18. Practice: If the mA = 58°, find the measure of an angle that makes a vertical angle with A.

  19. Practice: If the mA = 58°, find the measure of an angle that makes a linear pair with A.

  20. Practice: If the mA = 58°, find the measures of the angles formed when A is bisected.

  21. Homework pp. 137-141

  22. A F G E B D C ►A. Exercises mAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°. 7. Name two pairs of supplementary angles.

  23. A F G E B D C ►A. Exercises mAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°. 9. What is mFGE?

  24. ►B. Exercises Give the reason for each step in the proofs below. 18-22. Theorem 4.3 Angles that form a linear pair are supplementary. Given:PAB and BAQ form a linear pair Prove: PAD and BAQ are supplementary

  25. ■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be? 41. Addition property of 

  26. ■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be? 42. Multiplication property of 

  27. ■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be? 43. Reflexive property of 

  28. ■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be? 44. Transitive property of 

  29. ■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be? 45. Why is  not an equivalence relation?

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