# 5.6 Angle Bisectors and

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1 age 1 of ngle isectors and erpendicular isectors oal Use angle bisectors and perpendicular bisectors. ey Words distance from a point to a line equidistant angle bisector p. 61 perpendicular bisector The distance from a point to a line is measured by the length of the perpendicular segment from the point to the line. When a point is the same distance from one line as it is from another line, the point is equidistant from the two lines. TOR 5.3 ngle isector Theorem Words If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If 1 2 then 1 2 jogging path fountain 15 ft 15 ft bike path The fountain is equidistant from the jogging path and the bike path. ymbols If ma1 ma2, then &*c&*. X rove that TTWUcT VWU. iven UW &&( bisects atuv. TUTW and TUVW are right triangles. rove TTWUcTVWU. olution tatements 1 Use the ngle isector Theorem Reasons W T V U tudent elp TUY TI You can also show that the triangles in xample 1 are congruent by the ongruence Theorem. 1. UW &**( bisects atuv. 2.TUTW and TUVW are right triangles. 3. &***cwu 4. WV &**cwt 1. iven 2. iven 3. Reflexive rop. of ongruence 4. ngle isector Theorem 5. TTWU c TVWU 5. ongruence Theorem 5.6 ngle isectors and erpendicular isectors 273

2 age 2 of 8 erpendicular isectors segment, ray, or line that is perpendicular to a segment at its midpoint is called a perpendicular bisector. perpendicular bisector midpoint of &* TOR 5.4 erpendicular isector Theorem Words If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. ymbols If is on the perpendicular bisector of &*, then &*c &*. If then X 2 Use erpendicular isectors Use the diagram to find. olution In the diagram, ^&( is the perpendicular bisector of &*. y the erpendicular isector 8x 5x 12 Theorem,. 5x 12 8x 3x 12 ubtract 5x from each side. 3 x 1 2 ivide each side by x 4 implify. You are asked to find, not just the value of x. NWR 8x 8p4 32 Use ngle isectors and erpendicular isectors 1. ind. 2. ind. 3. ind. x 3 2x 1 4x x x 5 x hapter 5 ongruent Triangles

3 age 3 of 8 X In the diagram, N ^&*( is the perpendicular bisector of T&*. rove that TT is isosceles. olution To prove that TT is isosceles, show that T. tatements 3 Use the erpendicular isector Theorem 1. N ^&*( is the bisector of T&*. Reasons 1. iven 2. T 2. erpendicular isector Theorem 3. TT is isosceles. 3. ef. of isosceles triangle T N Intersecting isectors One consequence of the erpendicular isector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. = = areers IITI NNR y finding a location for a warehouse that is easily accessible to all its stores, a facilities planner helps a company save money and run more efficiently. areer inks Z O N. O X company plans to build a warehouse that is equidistant from each of its three stores,,, and. Where should the warehouse be built? olution Think of the stores as the vertices of a triangle. The point where the perpendicular bisectors intersect will be equidistant from each store. 1 Trace the location of the 2 raw the perpendicular stores on a piece of paper. bisectors of &*, &*, and &*. onnect the points of the abel the intersection of locations to form T. the bisectors. 4 Use Intersecting isectors of a Triangle tore tore tore NWR ecause is equidistant from each vertex of T, the warehouse should be built near location. 5.6 ngle isectors and erpendicular isectors 275

4 age 4 of xercises uided ractice Vocabulary heck omplete the statement. 1. If a point is on the bisector of an angle, then it is? from the two sides of the angle. 2. If is on the? of &*, then is equidistant from and. kill heck Use the information in the diagram to find the measure. 3. ind. 4. ind. 16 x 1 2x 1 5. ind. 6. ind QR. 5 5 R 5x 3x 8 12 ractice and pplications xtra ractice ee p Visualize It! opy each diagram on a piece of paper. Then draw a segment that represents the distance from to &* omework elp xample 1: xs. 32, 33 xample 2: xs xample 3: xs. 32, 33 xample 4: xs rror nalysis xplain why aige cannot make this conclusion, given the diagram shown. y the ngle isector Theorem, x 7. aige x hapter 5 ongruent Triangles

5 age 5 of 8 Using lgebra ind the value of x x 5 x x x Roof Trusses In the diagram of the roof truss shown below, you are given that &* bisects a and that a and a are right angles. What can you say about &* and &*? Why? Using isectors Use the diagram to find the indicated measure(s). 14. ind ma. 15. ind V. 16. ind. 38 V 18 U T ind. 18. ind Q. 19. ind and. areers x 2x 3 N R x 2 2x 1 3x x 6 IVI NINR plan and build large construction projects, such as bridges, canals, and tunnels. 20. ridges In the photo, the road is perpendicular to the support beam and &*c &*. What theorem allows you to conclude that &*c &*? xplain. areer inks Z O N. O 5.6 ngle isectors and erpendicular isectors 277

6 age 6 of 8 tudent elp OO or more about soccer, see p occer One way a goalie can determine a good defensive position is to imagine a triangle formed by the goal posts and the ball. 21. When the ball is far from the goal, the goalie most likely stands on line l. ow is l related to the goal line ( &*)? 22. s the ball moves closer, the goalie moves from line l to other places in front of the goal. ow should &*( relate to a? xplain. Using erpendicular isectors Use the information in the diagram. 23. ind and. 24. ind VR and VQ. 25. Name all congruent segments. T R 2 16 V U Itudent elp I Z O N. O OWOR xtra help with problem solving in xs is at classzone.com nalyzing a ap In xercises 26 29, use the map shown and the following information. city planner is trying to decide whether a new household at point X should be covered by fire station,, or. 26. Trace the points,,, and X on a piece of paper and draw the segments &*, &*, and &*. 27. raw the perpendicular bisectors of &*, &*, and &*. heck that they meet at a point. 28. The perpendicular bisectors divide the town into three regions. hade the region closest to fire station red. hade the region closest to fire station blue. hade the region closest to fire station gray. X 29. Writing In an emergency at household X, which fire station should respond? xplain your choice. 278 hapter 5 ongruent Triangles

7 age 7 of 8 Technology In xercises 30 and 31, use geometry drawing software to complete the steps below. 1 raw &*. ind the midpoint of &* and label it. 2 onstruct the perpendicular bisector of &* through. 3 onstruct point along the perpendicular bisector. onstruct &* and &*. 30. What is the relationship between &* and &*? easure &* and &* to verify your answer. 31. ove to another point along the perpendicular bisector. Will the relationship between &* and &* stay the same? Why? 32. roving the erpendicular isector Theorem ill in the missing statements and reasons. iven rove ^&( is the perpendicular bisector of &*. tatements Reasons 1. ^&( is the perpendicular bisector of &*. 2. &*c 1.? 2.? 3.? 3. lines form right angles. 4.? 4. Right angles are congruent. 5.? 5. Reflexive rop. of ongruence 6.TcT 7. &*c 6.? 7.? 8.? 8. ef. of congruent segments tudent elp OO or help with writing proofs, see p hallenge Use the diagram and the information below to prove the ngle isector Theorem. iven rove is on the bisector of a. &* &(, &** &( &*c &** int: irst prove that TcT. 5.6 ngle isectors and erpendicular isectors 279

8 age 8 of 8 tandardized Test ractice 34. ultiple hoice In the figure at the right, what is R? R x 2x ultiple hoice In the figure above, what is? ultiple hoice What can you say about the figure below, in which ^&( is the perpendicular bisector of &*? ll of these ixed Review Translations in a oordinate lane ind the image of the given point using the translation (x, y) (x 3, y 6). (esson 3.7) 37. (5, 1) 38. ( 2, 3) 39. ( 4, 4) 40. (0, 6) 41. (6, 2) 42. (2, 5) 43. (10, 12) 44. ( 1, 1) etermining ongruent Triangles What theorem or postulate, if any, can you use to show that the triangles are congruent? xplain your reasoning. (esson 5.5) N lgebra kills Ordering Numbers Write the numbers in order from least to greatest. (kills Review, p. 662) 48. 3, 3, 0.3, 0.3, 0.6, , 1, 0.75, 4, 1.25, , 0.1, 0, 4.0, 0.1, , 3.1, 3.8, 3.9, 3, , 1, 1.1, 1, 0.5, 0.1, , 1, 2.1, 3.25, 2.5, 5 olving quations olve the equation. (kills Review, p. 673) 54. 4x y d a 9a x 2 3x r 2 5r q 2q z 5 4z t 10 12t 280 hapter 5 ongruent Triangles

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