1.2 Perpendicular Lines
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1 Name lass ate 1.2 erpendicular Lines Essential Question: What are the key ideas about perpendicular bisectors of a segment? 1 Explore onstructing erpendicular isectors and erpendicular Lines You can construct geometric figures without using measurement tools like a ruler or a protractor. y using geometric relationships and a compass and a straightedge, you can construct geometric figures with greater precision than figures drawn with standard measurement tools. In Steps, construct the perpendicular bisector of. Resource Locker lace the point of the compass at point. Using a compass setting that is greater than half the length of, draw an arc. Without adjusting the compass, place the point of the compass at point and draw an arc intersecting the first arc in two places. Label the points of intersection and. Use a straightedge to draw, which is the perpendicular bisector of. Houghton Mifflin Harcourt ublishing ompany In Steps E, construct a line perpendicular to a line l that passes through some point that is not on l. lace the point of the compass at. raw an arc that intersects line l at two points, and. l E Use the methods in Steps to construct the perpendicular bisector of. l l ecause it is the perpendicular bisector of, then the constructed line through is perpendicular to line l. Module 1 15 Lesson 2
2 Reflect 1. In Step of the first construction, why do you open the compass to a setting that is greater than half the length of? 2. What If? Suppose Q is a point on line l. Is the construction of a line perpendicular to l through Q any different than constructing a perpendicular line through a point not on the line, as in Steps and E? Explain 1 roving the erpendicular isector Theorem Using Reflections You can use reflections and their properties to prove a theorem about perpendicular bisectors. These theorems will be useful in proofs later on. erpendicular isector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Example 1 rove the erpendicular isector Theorem. Given: is on the perpendicular bisector m of. rove: = m onsider the reflection across. Then the reflection Reflect of point across line m is also on lso, the reflection of of. Therefore, = because, which is the line of reflection. 3. iscussion What conclusion can you make about KLJ in the diagram using the erpendicular isector Theorem? K M L because point lies across line m is by the definition preserves distance. Houghton Mifflin Harcourt ublishing ompany J Module 1 16 Lesson 2
3 Your Turn Use the diagram shown. is the perpendicular bisector of. 4. Suppose E = 16 cm and = 20 cm. Find. E Explain 2 5. Suppose E = 15 cm and = 25 cm. Find. roving the onverse of the erpendicular isector Theorem The converse of the erpendicular isector Theorem is also true. In order to prove the converse, you will use an indirect proof and the ythagorean Theorem. In an indirect proof, you assume that the statement you are trying to prove is false. Then you use logic to lead to a contradiction of given information, a definition, a postulate, or a previously proven theorem. You can then conclude that the assumption was false and the original statement is true. a c a 2 + b 2 = c 2 Recall that the ythagorean Theorem states that for a right triangle with legs of length a and b and a hypotenuse of length c, a 2 + b 2 = c 2. b onverse of the erpendicular isector Theorem If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Houghton Mifflin Harcourt ublishing ompany Example 2 rove the onverse of the erpendicular isector Theorem Given: = rove: is on the perpendicular bisector m of. Step : ssume what you are trying to prove is false. ssume that is not on the perpendicular bisector m of. Then, when you draw a perpendicular line from to the line containing and, it intersects at point Q, which is not the of. Step : omplete the following to show that this assumption leads to a contradiction. Q forms two right triangles, Q and. So, Q 2 + Q 2 = 2 and Q 2 + Q 2 = by the Theorem. Subtract these equations: Q 2 + Q 2 = Q + Q = 2 Q Q = However, 2-2 = 0 because. Therefore, Q Q = 0. This means that Q 2 2 = Q and Q = Q. This contradicts the fact that Q is not the midpoint of. Thus, the initial assumption must be incorrect, and must lie on the of. m Q Module 1 17 Lesson 2
4 Reflect 6. In the proof, once you know Q 2 = Q 2, why can you conclude that Q = Q? Your Turn 7. is 10 inches long. is 6 inches long. Find the length of. Explain 3 roving Theorems about Right ngles The symbol means that two figures are perpendicular. For example, l m or XY. Example 3 rove each theorem about right angles. If two lines intersect to form one right angle, then they are perpendicular and they intersect to form four right angles. Given: m 1 = 90 rove: m 2 = 90, m 3 = 90, m 4 = Statement 1. m 1 = Given 2. 1 and 2 are a linear pair. 2. Given 3. 1 and 2 are supplementary. 3. Linear air Theorem Reason 4. m 1 + m 2 = efinition of supplementary angles m 2 = Substitution roperty of Equality 6. m 2 = Subtraction roperty of Equality 7. m 2 = m 4 7. Vertical ngles Theorem 8. m 4 = Substitution roperty of Equality 9. m 1 = m 3 9. Vertical ngles Theorem 10. m 3 = Substitution roperty of Equality If two intersecting lines form a linear pair of angles with equal measures, then the lines are perpendicular. Given: m 1 = m 2 l 1 2 m rove: l m y the diagram, 1 and 2 form a linear pair so 1 and 2 are supplementary by the. y the definition of supplementary angles, m 1 + m 2 =. It is also given that, so m 1 + m 1 = 180 by the. dding gives 2 m 1 = 180 and m 1 = 90 by the ivision roperty of Equality. Therefore, 1 is a right angle and l m by the. Houghton Mifflin Harcourt ublishing ompany Module 1 18 Lesson 2
5 Reflect 8. State the converse of the theorem in art. Is the converse true? Your Turn 9. Given: b ǁ d, c ǁ e, m 1 = 50, and m 5 = 90. Use the diagram to find m 4. b a 1 d c e Elaborate 10. iscussion Explain how the converse of the erpendicular isector Theorem justifies the compass-and-straightedge construction of the perpendicular bisector of a segment. Houghton Mifflin Harcourt ublishing ompany 11. Essential Question heck-in How can you construct perpendicular lines and prove theorems about perpendicular bisectors? Module 1 19 Lesson 2
6 Evaluate: Homework and ractice 1. How can you construct a line perpendicular to line l that passes through point using paper folding? l Online Homework Hints and Help Extra ractice 2. heck for Reasonableness How can you use a ruler and a protractor to check the construction in Elaborate Exercise 11? 3. escribe the point on the perpendicular bisector of a segment that is closest to the endpoints of the segment. 4. Represent Real-World roblems field of soybeans is watered by a rotating irrigation system. The watering arm,, rotates around its center point. To show the area of the crop of soybeans that will be watered, construct a circle with diameter. Use the diagram to find the lengths. is the perpendicular bisector of. Q is the perpendicular bisector of. = =. 5. Suppose = 5 cm. What is the length of? 6. Suppose = 5 cm and Q = 8 cm. What is the length of Q? Q Houghton Mifflin Harcourt ublishing ompany Image redits: sima/ Shutterstock Module 1 20 Lesson 2
7 7. Suppose = 12 cm and Q = 10 cm. What is the length of Q? 8. Suppose = 3 cm and = 12 cm. What is the length of? Given: = and =. Use the diagram to find the lengths or angle measures described. 9. Suppose m 2 = 38. Find m Suppose = 10 cm and = 6 cm. What is the length of? 11. Find m 3 + m 4. Given: m ǁ n, x ǁ y, and y m. Use the diagram to find the angle measures. Houghton Mifflin Harcourt ublishing ompany x y m Suppose m 7 = 30. Find m Suppose m 1 = 90. What is m 2 + m 3 + m 5 + m 6? n a Module 1 21 Lesson 2
8 Use this diagram of trusses for a railroad bridge in Exercise Suppose E is the perpendicular bisector of F. Which of the following statements do you know are true? Select all that apply. Explain your reasoning.. = F. m 1 + m 2 = 90. E is the midpoint of F.. m 3 + m 4 = 90 E. E F lgebra Two lines intersect to form a linear pair with equal measures. One angle has the measure 2x and the other angle has the measure (20y - 10). Find the values of x and y. Explain your reasoning. 16. lgebra Two lines intersect to form a linear pair of congruent angles. The measure of one angle is (8x + 10) and the measure of the other angle is 15y ( 2 ). Find the values of x and y. Explain your reasoning. H.O.T. Focus on Higher Order Thinking 17. ommunicate Mathematical Ideas The valve pistons on a trumpet are all perpendicular to the lead pipe. Explain why the valve pistons must be parallel to each other. lead pipe Houghton Mifflin Harcourt ublishing ompany valve pistons Module 1 22 Lesson 2
9 18. Justify Reasoning rove the theorem: In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Given: RS and rove: RS Statements Reasons R T V S 19. nalyze Mathematical Relationships omplete the indirect proof to show that two supplementary angles cannot both be obtuse angles. Given: 1 and 2 are supplementary. rove: 1 and 2 cannot both be obtuse. ssume that two supplementary angles can both be obtuse angles. So, assume that 1 and 2. Then m 1 > 90 and m 2 > by m 1 + m 2 > information.. dding the two inequalities,. However, by the definition of supplementary angles,. So m 1 + m 2 > 180 contradicts the given This means the assumption is, and therefore Houghton Mifflin Harcourt ublishing ompany. Module 1 23 Lesson 2
10 Lesson erformance Task utility company wants to build a wind farm to provide electricity to the towns of cton, axter, and oleville. ecause of concerns about noise from the turbines, the residents of all three towns do not want the wind farm built close to where they live. The company comes to an agreement with the residents to build the wind farm at a location that is equally distant from all three towns. cton 1.5 in. 120 oleville 4 in. Scale 1 in. : 10 mi axter a. Use the drawing to draw a diagram of the locations of the towns using a scale of 1 in. : 10 mi. raw the 4-inch and 1.5-inch lines with a 120 angle between them. Write the actual distances between the towns on your diagram. b. Estimate where you think the wind farm will be located. c. Use what you have learned in this lesson to find the exact location of the wind farm. What is the approximate distance from the wind farm to each of the three towns? Houghton Mifflin Harcourt ublishing ompany Module 1 24 Lesson 2
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