Slide 1 / 165 Slide 2 / 165 New Jersey Center for Teaching and Learning Geometry Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of Triangles students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning 2014-02-12 community, and/or provide access to course materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 165 Slide 4 / 165 Triangles Triangles Click on a title to Table Of Contents Table Of Contents link to that page. Labs Perpendiculars And Bisectors Of Segments And Angles · Perpendicular Bisectors Of A Segment · Perpendicular Bisector Theorem Lab Angle Bisectors · · Bisectors Of A Triangle · Angle Bisector Theorem Lab · Click on a title to Medians Of A Triangle · link to that page. Altitudes Of A Triangle · Perpendicular Bisectors of a Triangle Labs · Points Of Concurrency · Angle Bisectors of a Triangle Labs · Midsegments Of A Triangle · Inequalities In One Triangle · Medians of a Triangle Labs · Hinge Theorem · Altitudes of a Triangle Lab · Indirect Proof · Special Cases Lab & Sketch · Midsegments of a Triangle Labs · Inequalities in One Triangle Lab & Sketch · Triangle Inequality Lab & Sketch · Slide 5 / 165 Slide 6 / 165 Vocabulary Review Match the correct term for the red line in the sketch on the left with the terms on the right. Perpendiculars and Bisectors Answer A. Bisector 2. 1. of Segments and Angles B. Perpendicular C. Perpendicular Bisector D. Angle Bisector 4. 3. Return to Table of Contents
Slide 7 / 165 Slide 8 / 165 Term Definition Diagram Term Definition Diagram A segment, ray, line or Bisector of a plane that divides a segment into 2 equal segment A ray that divides parts. and angle into 2 Angle Bisector adjacent, congruent angles. A segment, ray, line or Perpendicular to plane that intersects a a segment segment at right angles. A segment, ray, line, or Perpendicular plane that is Bisector of a perpendicular to a segment segment at its midpoint. Slide 9 / 165 Slide 10 / 165 1 A bisector of a segment is also a perpendicular bisector 2 If AB is the bisector of XY then XM is 7. of a segment. True True False Answer Answer False Slide 11 / 165 Slide 12 / 165 3 HJ is the angle bisector of AHB, find the measure of AHB. 4 The perpendicular bisector of a segment intersects the segment at its _____. A 75 0 0 A endpoint B 21 B midpoint 0 C 150 C top D none of these Answer Answer D bottom
Slide 13 / 165 Slide 14 / 165 The Perpendicular Bisector Theorem To investigate the perpendicular bisector theorem go to the Perpendicular Bisectors lab titled, "The Perpendicular Bisector Theorem Lab." of a Segment Go to the Click here to review "Perpendicular Bisector constructing a perpendicular Theorem Lab" bisector of a segment. "math is fun" Return to table of contents Slide 15 / 165 Slide 16 / 165 Fill in the blanks according to the The Perpendicular Bisector Theorem diagram below. A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. CD is the _____ of AB therefore AC ≅ _____ and BC = _____ ? ? ? Answer XM is the perpendicular bisector of AB therefore XA ≅ XB Slide 17 / 165 Slide 18 / 165 Discuss Why JM ≅ LM Using ≅ If JM ML, and K is the midpoint of JL is it possible to prove M lies on the Congruent Triangles. perpendicular bisector of JL? Given: KM is the perpendicular bisector of JL. ≅ Prove: JM LM Answer Hint: Construct Answer segment KM. Remember when proving the perpendicular bisector theorem Discuss a plan for the proof of the Converse of the you may not use it as a reason in Perpendicular Bisector Theorem. your proof.
Slide 19 / 165 Slide 20 / 165 Fill in the blanks according to the The Converse of the Perpendicular diagram below. Bisector Theorem PS ≅ therefore S lies on the of PR and PQ ≅ ? ? ? If a point is equidistant from the endpoints of a segment then it lies on the perpendicular bisector of the segment. Answer ≅ If JM ML, then M lies on the perpendicular bisector of JL. Slide 21 / 165 Slide 22 / 165 Emani and Jada both live in the same town on the same street, Using the diagram below, find JK, KL Main Street. Emani lives on the corner of Central and Main and and KM. Jada lives on the corner of Prospect and Main. Their friend Karen lives on Park Place and Harrison Street . Karen lives the same distance from both Emani and Jada. What conclusion can you make about the relationship between Park Place and Main Street? Answer Answer Harrison Street Harrison Street Prospect Street Central Avenue Park Place Karen Main Street Main Street Emani J ada Slide 23 / 165 Slide 24 / 165 5 Tell whether the information in the diagram allows you to 6 Tell whether the information in the diagram allows you to conclude that X is on the perpendicular bisector of PQ. conclude that X is on the perpendicular bisector of PQ. Explain your reasoning. Yes Yes Answer No Answer No
Slide 25 / 165 Slide 26 / 165 7 In the diagram, XY AB and AY YB. Find YB and XA. ≅ 8 In the diagram XY is the perpendicular bisector of AB. Because AZ=BZ=15, what can you conclude about point Z? A 9, 40 A ZY = 15 B 4.5, 40 B Z is on XY, the perpendicular bisector of AB. C 4.5, 41 Answer Answer C ZY = 9 D 9, 41 D No conclusion can be drawn. Slide 27 / 165 Slide 28 / 165 The Angle Bisector Theorem To investigate the angle bisector theorem go to the lab titled, "The Angle Bisector Theorem Lab." Angle Bisectors Go to the Click here to review "Angle Bisector constructing an angle Theorem Lab" bisector. "math open reference" Return to Table of Contents Slide 29 / 165 Slide 30 / 165 Fill in the blanks according to the The Angle Bisector Theorem diagram below. A point on the angle bisector is equidistant from the sides ? EG is the _____ of <FEH and GF is _____ to EF and GH is ? of the angle. _____ to EH therefore FG ≅ _____. ? ? Answer BX is the angle bisector of <ABC therefore XY ≅ XZ and <XYB and <XZB are right angles.
Slide 31 / 165 Slide 32 / 165 If X lies on the interior of ABC and is Tell whether the information in the equidistant from the sides of ABC does diagram allows you to conclude that A is point X lie on the angle bisector of ABC? on the bisector of Y. Answer Hint: Construct BX Answer Remember: For a point to be equidistant to a ray it must be the perpendicular distance. Discuss a plan for the proof of the converse of the angle bisector theorem. Slide 33 / 165 Slide 34 / 165 Fill in the blanks according to the Converse of the Angle Bisector Theorem diagram below. If a point lies on the interior of an angle and is equidistant from the sides of the angle then the point lies on the angles ? ? ? EH _____EF and GH ______GF also EH _____ GH therefore H is bisector of the angle. ? ? _____ from the sides of EFG and lies on the _____ of EFG. If X is on the interior of ABC and equidistant from the sides of ABC then X lies on the angle bisector of Answer ABC. Slide 35 / 165 Slide 36 / 165 Tell whether the information in the 9 Is there enough information to conclude whether or not diagram allows you to conclude that J point D is on the bisector of B? lies on the angle bisector of KLM. Yes No Answer Answer
Slide 37 / 165 Slide 38 / 165 10 Is there enough information to conclude whether or not 11 In the diagram, AB bisects <DAF, BE AE, BG AG, and point D is on the bisector of B? BE = 2. Find BG. A 2 Yes Answer B 3 C 6 No Answer D None of the above Slide 39 / 165 Slide 40 / 165 12 In the diagram, AB bisects DAF, CD AD, CF AF, and CD=CF=6. What can you conclude about point C? Bisectors of a Triangle A No conclusion can be drawn. There are Two Types: B C lies on the bisector of DAF. Answer C C does not lie on the bisector of DAF. 1. Perpendicular Bisectors of a Triangle D C lies on the bisector of AD. 2. Angle Bisectors of a Triangle Return to Table of Contents Slide 41 / 165 Slide 42 / 165 Concurrency of Perpendicular Perpendicular Bisectors of a Triangle Bisectors of a Triangle Theorem To investigate the perpendicular bisectors of a triangle go The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. to the lab titled, "The Perpendicular Bisectors of a Triangle Lab." Go to the Go to the "Concurrency of Perpendicular "Concurrency of Perpendicular Bisectors of a Triangle Lab" using Bisectors of a Triangle Lab" using gsp. paper-folding.
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