3.2 - Theorems About Perpendicular Lines

1. 3.2 - Theorems About Perpendicular Lines Objectives: #3, #4 HW #2

2. Activity – Intersecting Lines

3. Activity – Intersecting Lines

5. Example 1 In the diagram, r sand r t. Determine whether enough information is given to conclude that the statement is true. Explain your reasoning. a. 3  5 b. 4  5 c. 2  3 SOLUTION a. Yes, enough information is given. Both angles are right angles. By Theorem 3.1, they are congruent. Perpendicular Lines and Reasoning

6. Example 1 b. Yes, enough information is given. Lines rand tare perpendicular. So, by Theorem 3.2, 4 is a right angle. By Theorem 3.1, all right angles are congruent. c. Not enough information is given to conclude that 2  3. Perpendicular Lines and Reasoning

7. Checkpoint In the diagram, g eand g f. Determine whether enough information is given to conclude that the statement is true. Explain. 1. 6  10 Yes; 6 and 10 are right angles and all right angles are congruent. ANSWER 2. 7  10 Yes; since g e, 7 is a right angle; 10 is also a right angle; all right angles are congruent. ANSWER 3. 6  8 no ANSWER Perpendicular Lines and Reasoning

8. Checkpoint In the diagram, g eand g f. Determine whether enough information is given to conclude that the statement is true. Explain. 4. 7  11 Yes; since g e, 7 is a right angle; since g  f, 11 is a right angle; all right angles are congruent. ANSWER 5. 7  9 no. ANSWER 6. 6  11 Yes; 6 is a right angle; since g f, 11 is a right angle; all right angles are congruent. ANSWER Perpendicular Lines and Reasoning

10. Example 3 In the diagram at the right, EF EH and mGEH=30°. Find the value of y. SOLUTION FEGand GEHare adjacent acute angles and EF EH. So, FEGand GEHare complementary (Theorem 3.4). 6y° + 30° = 90° mFEG + mGEH = 90° 6y= 60 Subtract 30 from each side. y= 10 Divide each side by 6. The value of yis 10. ANSWER Use Algebra with Perpendicular Lines

11. Checkpoint Find the value of the variable. Explain your reasoning. 7. EFG HFG ANSWER EH and FGintersect to form adjacent congruent angles, so the lines are perpendicular. Perpendicular lines intersect to form 4 right angles, so mEFG= 90°. 5x = 90; x = 18. Use Algebra with Perpendicular lines

12. Checkpoint 8. AB AD BACand CADare adjacent acute angles and ABAD, so BACand CADare complementary. 36° + 9y° = 90°; 9y = 54; y = 6. ANSWER Use Algebra with Perpendicular lines Find the value of the variable. Explain your reasoning.

13. Checkpoint 9. KJKL,JKM MKL JKMand MKLare adjacent acute angles and KJKL, so JKMand MKLare complementary. z°+ z°= 90°; 2z =90;z =45. ANSWER Use Algebra with Perpendicular lines Find the value of the variable. Explain your reasoning.

14. Example 2 In the helicopter at the right, are AXBand CXBright angles? Explain. SOLUTION If two lines intersect to form adjacent congruent angles, as AC and BD do, then the lines are perpendicular (Theorem 3.3). So, ACBD. Because AC and BD are perpendicular, they form four right angles (Theorem 3.2). So, AXBand CXBare right angles. Use Theorems About Perpendicular Lines