Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Theorems About Perpendicular Lines Section 3.2 Objective • Use theorems about perpendicular lines. Theorems • 3.1 -3.4 Perpendicular Line Theorems Key Vocabulary (Review) • Complementary Angles – two angles with measures that have a sum of 90˚. • Perpendicular Lines – two lines that intersect to form right angles. Theorem 3.1 All right angles are congruent. If m∠A = 90o and m∠B = 90o, then ∠A ≅ ∠B. B A Theorem 3.2 If two lines are perpendicular, then they intersect to form four right angles If n ⊥ m, then m∠1 = 90o, m∠2 = 90o, m∠3 = 90o, and m∠4 = 90o. 1 4 2 3 n m Marking Right Angles • Theorem 3.2 tells us that if one right angle is marked on a pair of intersecting lines, then the other three angles are also right angles. • Do not need to mark all four right angles when two lines intersect. Marking only one right angle is sufficient. Example 1 In the diagram, r s and r t. Determine whether enough information is given to conclude that the statement is true. Explain your reasoning. a. 3 5 a. Yes, enough information is given. Both angles are right angles. By Theorem 3.1, they are congruent. Example 1 b. 4 5 b. Yes, enough information is given. Lines r and t are perpendicular. So, by Theorem 3.2, 4 is a right angle. By Theorem 3.1, all right angles are congruent. c. 2 3 c. Not enough information is given to conclude that 2 3. Your Turn: In the diagram, g e and g f. Determine whether enough information is given to conclude that the statement is true. Explain. 1. 6 10 ANSWER Yes; 6 and 10 are right angles and all right angles are congruent. 2. 7 10 ANSWER Yes; since g e, 7 is a right angle; 10 is also a right angle; all right angles are congruent. 3. 6 8 ANSWER no Your Turn: In the diagram, g e and g f. Determine whether enough information is given to conclude that the statement is true. Explain. 4. 7 11 ANSWER Yes; since g e, 7 is a right angle; since g f, 11 is a right angle; all right angles are congruent. 5. 7 9 ANSWER no. 6. 6 11 ANSWER Yes; 6 is a right angle; since g f, 11 is a right angle; all right angles are congruent. Theorem 3.3 If two lines intersect to form adjacent congruent angles, then the lines are perpendicular. If ∠1 ≅ ∠2, then AC ⊥ BD. B A 1 2 D C Theorem 3.4 If two sides of adjacent acute angles are perpendicular, then the angles are complementary. If EF ⊥ EH, then m∠3 + m∠4 = 90o. F G 3 4 E H Example 2 In the helicopter at the right, are AXB and CXB right 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, AXB and CXB are right angles. Example 3 In the diagram at the right, EF EH and mGEH = 30°. Find the value of y. SOLUTION FEG and GEH are adjacent acute angles and EF EH. So, FEG and GEH are complementary (Theorem 3.4). 6y° + 30° = 90° 6y = 60 y = 10 ANSWER mFEG + mGEH = 90° Subtract 30 from each side. Divide each side by 6. The value of y is 10. Your Turn: Find the value of the variable. Explain your reasoning. 1. EFG HFG ANSWER EH and FG intersect to form adjacent congruent angles, so the lines are perpendicular. Perpendicular lines intersect to form 4 right angles, so mEFG = 90°. 5x = 90; x = 18. Your Turn: Find the value of the variable. Explain your reasoning. 2. AB AD ANSWER BAC and CAD are adjacent acute angles and AB AD, so BAC and CAD are complementary. 36° + 9y° = 90°; 9y = 54; y = 6. Your Turn: Find the value of the variable. Explain your reasoning. 3. KJ KL, JKM MKL ANSWER JKM and MKL are adjacent acute angles and KJ KL, so JKM and MKL are complementary. z° + z° = 90°; 2z = 90; z = 45. PRACTICE PROBLEMS GV AE F A 5 6 G B V 1 2 8 3 C 4 E AE FV F A 5 6 G B V 1 2 8 3 C 4 E 4 1 F A 5 6 G B V 1 2 8 3 C 4 E 3 4 F A 5 6 G B V 1 2 8 3 C 4 E 1 2 90 F A 5 6 G B V 1 2 8 3 C 4 E GVA is a right F A 5 6 G B V 1 2 8 3 C 4 E 6 & 3 are supplementary F A 5 6 G B V 1 2 8 3 C 4 E 6 & 2 are complementary F A 5 6 G B V 1 2 8 3 C 4 E Joke Time • Why do mother kangaroos hate rainy days? • Because the kids have to play inside. • Why did the chicken go to the middle of the road? • To lay it on the line. • Why did the chicken cross the basketball court to talk with the ref? • Because he was calling all fowls! Assignment • Section 3.2, pg. 117-120: #1-27 odd, 3143 odd