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Understanding Perpendicular Lines: Definitions, Theorems, and Examples in Geometry

This chapter delves into the concept of perpendicular lines, defined as two lines that intersect to form right angles (90 degrees). It introduces the symbol for perpendicularity (┴) and provides essential theorems regarding adjacent angles. Topics covered include the formation of complementary angles and proofs related to perpendicularity. Examples illustrate angle measures and calculations, such as solving for unknowns in angle relationships. Additionally, exercises are provided to reinforce learning and practice deductive reasoning in geometry.

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Understanding Perpendicular Lines: Definitions, Theorems, and Examples in Geometry

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  1. CHAPTER 2: DEDUCTIVE REASONING Section 2-5: Perpendicular Lines

  2. PERPENDICULAR LINES Definition Perpendicular Lines: 2 lines that intersect to form right angles (90 degree angles). ┴is the symbol for perpendicular l┴mThis is read: “line l is perpendicular to line m” m l

  3. EXAMPLE If m 1 = 90 , find the measure of each angle. m 2 = 90 m 3 = 90 m 4 = 90 2 1 3 4

  4. THEOREMS Theorem 2-4: If lines are ┴, they form ≡ adjacent angles. Theorem 2-5: If 2 lines form ≡ adjacent angles, they are ┴. Theorem 2-6: If the non-common sides of 2 adjacent angles are ┴, then the angles are complementary.

  5. OA ┴ OC m AOC = 90 m 1 + m 2 = m AOC m 1 + m 2 = 90 1 and 2 are complementary Given Def. of ┴ lines Angle Add. Post. Substitution Def. of complementary angles PROOF A B Given: OA ┴ OC Prove: 1 and 2 are complementary 1 C 2 O

  6. m EYO m OYM m MYT m EYT 90 – 30 = 60 90 – 60 = 30 90 – 30 = 60 60 + 90 = 150 EXAMPLE If m GYE = 30, find each angle measure YE ┴ YM, YO ┴ GT O M E G T Y

  7. EXAMPLE m EYO = 3x, m OYM = 2x + 15 Find x. 3x + 2x + 15 = 90 5x + 15 = 90 5x = 75 x = 15 O M E G T

  8. EXAMPLE m GYE = x - 2, m EYO = 3x + 12 Find x. x – 2 + 3x + 12 = 90 4x + 10 = 90 4x = 80 x = 20 O M E G T

  9. CLASSWORK/HOMEWORK • CLASSWORK Pg. 57: Classroom Exercises 1, 2-10 even • HOMEWORK Pg. 58: Written Exercises 2-24 even

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