Proof EXAMPLE EXAMPLE. Given:

Transcription

1 4-7 hat ou ll earn o identify congruent overlapping triangles o prove two triangles congruent by first proving two other triangles congruent... nd hy o identify overlapping triangles in scaffolding, as in xample sing orresponding arts of ongruent riangles heck kills ou ll Need O for elp. ow many triangles will the next two figures in this pattern have? 5; 3 essons - and 4-3. an you conclude that the triangles are congruent? xplain. a. # and # b. # and #N c. # and #N yes; yes; rans. yes; rop. of O N 4-7. lan Objectives o identify congruent overlapping triangles o prove two triangles congruent by first proving two other triangles congruent xamples Identifying ommon arts sing ommon arts 3 sing wo airs of riangles 4 eparating Overlapping riangles sing Overlapping riangles in roofs ocabulary ip Overlapping triangles share part or all of one or more sides. ome triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a common side or angle. ou can simplify your work with overlapping triangles by separating and redrawing the triangles. eparate and redraw # and #. Identify the common angle. Identifying ommon arts ommon angle ath ackground he use of in overlapping triangles is fundamental to the investigation of quadrilaterals. or example, the proof that the diagonals of a rectangle are congruent follows easily from using the ostulate to prove that the overlapping right triangles formed by the diagonals are congruent. ore ath ackground: p. 96 uick heck ngineering he diagram at the left shows triangles from the scaffolding that workers used when they repaired and cleaned the tatue of iberty. a. Name the common side in # and #. b. Name another pair of triangles that share a common side. Name the common side. nswers may vary. ample: k and k; esson lanning and esources ee p. 96 for a list of the resources that support this lesson. oweroint ell inger ractice heck kills ou ll Need or intervention, direct students to: In overlapping triangles, a common side or angle is congruent to itself by the eflexive roperty of ongruence. esson 4-7 sing orresponding arts of ongruent riangles 4 lanning a roof esson 4-3: xample 3 xtra kills, ord roblems, roof ractice, h. 4 sing the heorem esson 4-6: xamples and 3 xtra kills, ord roblems, roof ractice, h. 4 pecial Needs or xample 3, help students recognize that they cannot prove unless they can first prove. y proving, students identify other pairs of congruent parts. learning style: visual elow evel se separable transparencies on an overhead projector and different-colored pens to help students distinguish overlapping triangles and congruent corresponding parts. learning style: visual 4

2 . each uided Instruction eaching ip arking the congruent parts of triangles is difficult when triangles overlap. y redrawing separate triangles, the congruent parts can be marked more easily. eaching ip oth plan and proof are given to help students focus on the gradual development of a proof. ath ip edrawing overlapping triangles can clarify relationships or make them even more confusing, depending on which overlapping triangles are redrawn. oint out that students may have to try several ideas before they find a good proof plan. oweroint dditional xamples Name the parts of their sides that and share in xample. and rite a lan for roof for xample that does not use overlapping triangles. abel the intersection of and point. O by if k Ok. how this congruence by... k O k (iven). l O l () roof 3. O (If base ' are O, the opp. sides are O.) uick heck roof uick heck 3 iven: sing ommon arts & > &, & > & rite a plan and then a proof to show that the two outside segments are congruent. rove: > lan: irst, separate the overlapping triangles. > by if # > #. how this congruence by. roof: iven eflexive rop. of iven rite a plan and then a proof. iven: # > # rove: > ee left. sing wo airs of ongruent riangles ometimes you can prove one pair of triangles congruent and then use their congruent corresponding parts to prove another pair congruent. sing wo airs of riangles iven: In the quilt, is the midpoint of and. rove: # > # rite a plan and then a proof. lan: # > # by if & > &. hese angles are congruent by if # > #. hese triangles are congruent by. roof: is the midpoint of and, so > and >. & > & because vertical angles are congruent. herefore, # > # by. & > & by, and & > & because they are vertical angles. herefore, # > # by. 3 rite a plan and then a proof. iven: >, & > & rove: # > # ee back of book. ostulate 4 hapter 4 ongruent riangles dvanced earners 4 ave students copy the diagram in xample, drawing. hen have them prove that and are parallel. nglish anguage earners ome students may not understand the term overlapping. se an overhead projector and transparencies with overlays to illustrate its meaning. 4 learning style: verbal learning style: visual

3 eal-orld roof uick heck xample (page 4) 4 4 hen triangles overlap, you can keep track of information by drawing other diagrams that separate the overlapping triangles. eparating Overlapping riangles iven: >, > rite a plan and then a proof to show that two small segments inside the triangle are congruent. rove: > lan: > by if # > #. his congruence holds by if & > &. hese are congruent by in # and #, which are congruent by. roof: tatements easons. >. iven. >. iven 3. & > & 3. Isosceles riangle heorem 4. > 4. eflexive roperty of ongruence 5. # > # & > & & > & 7. ertical angles are congruent. 8. # > # > 9. lan a proof. eparate the overlapping triangles in your plan. hen follow your plan and write a proof. ee margin. iven: & > &, & > & rove: > I or more exercises, see xtra kill, ord roblem, and roof ractice. ractice and roblem olving onnection he apanese paper-folding art of origami involves many overlapping triangles. O ractice by xample for elp In each diagram, the red and blue triangles are congruent. Identify their common side or angle.. l. 3. N uided Instruction 3 rror revention tudents may think they can prove directly from the information given. iscuss how proving acts as a bridge from the iven to proving. oint out that a proof often involves finding such a bridge between ideas. oweroint dditional xamples 3 rite a paragraph proof. iven:, & and & are right angles. rove: O (iven), l Ol (right angles) and O (eflexive rop.), so k Ok by. l Ol by, l Ol (vert. angles are O), and O (iven), so k Ok by. 4 se the iven from xample 4 to write a two-column proof to show that & &.. l Ol (eflexive). O, O (iven) 3. (ubtraction rop. of quality) 4., (eg. dd. ost.) 5. (ubstitution) 6. O (ef. of O) 7. k Ok () 8. l Ol () esources aily Notetaking uide aily Notetaking uide 4-7 dapted Instruction uick heck 4.. l O l; l O l (iven). O (eflexive rop. of O) esson 4-7 sing orresponding arts of ongruent riangles k O k () 4. O () 5. l O l (ert. ' are O.) 6. k O k () 7. O () losure xplain how can be used in the middle of a proof. ometimes you can prove a pair of triangles congruent and then use to prove another pair congruent. 43

4 3. ractice ssignment uide -9, -5 0,, 6- hallenge 3-5 est rep 6-30 ixed eview 3-40 omework uick heck o check students understanding of key skills and concepts, go over xercises 6, 0, 4, 9,. rror revention! xercise 7 In step b, identifying & using two different names may confuse students and prevent them from realizing that the eflexive roperty of ongruence applies. sk: hy is & named in two ways? to show the order of the corresponding vertices in k and k xercise 6 tudents need to recognize that because the same quantity, is being added to the congruent segments and. nrichment uided roblem olving eteaching dapted ractice ractice Name lass ate ractice 4-7 sing orresponding arts of ongruent riangles Name a pair of overlapping congruent triangles in each diagram. tate whether the triangles are congruent by,,,, or.. iven:,. iven:, 3. iven: 6, 6, and are right s O 4. iven:, 5. iven:, 6. iven:, ON, N O eparate and redraw the indicated triangles. Identify any common angles or sides O 4 6. ee back of book. xample (page 4) xamples 3, 4 (pages 4 and 43) pply our kills nline omework elp isit: chool.com eb ode: aue-0407 eparate and redraw the indicated triangles. Identify any common angles or sides. 4. # and # 5. # and # 6. # and # 7. eveloping roof omplete the flow proof. iven: & > &, > rove: & > & iven a.? eflexive rop. of O b.? roof 44 hapter 4 ongruent riangles c.? iven rite a plan and then a proof. 8. iven: >, > 9. iven: >, &>& rove: # > # rove: # > # 8 9. ee back of book. 0. iven: & > &, &3 > &4. iven: >, rove: # > # is the midpoint of. 0. ee back rove: # > # of book. 3 4 d.? e.? 5. nswers may vary. Open-nded raw the diagram described. amples are given.. raw a vertical segment on your paper. On the right side of the segment draw two triangles that share the given segment as a common side. ee left. 3. raw two triangles that have a common angle. ee left. 4. raw two regular pentagons, each with its five diagonals. a b. ee margin. a. In one, shade two triangles that share a common angle. b. In the other, shade two triangles that share a common side. 5. raw two regular hexagons and their diagonals. or these diagrams, do parts (a) and (b) of the preceding exercise. ee margin. O earson ducation, Inc. ll rights reserved. 7. and 8. and 9. and N N rite a two-column proof, a paragraph proof, or a flow proof. 0. iven:, #, #. iven:, rove: rove: 4. a. b. 5. a. b. 44

5 7.. O ; O (iven). l O l (eflexive rop. of O) 3. k O k () 4. l O l () roof 6. ultiple hoice In the diagram, & > &, >, and >. hich two triangles can you prove congruent by? # > # # > # # > # none of these 7. iven: >, > 8. iven: ', bisects, rove: & > & ee left. bisects &. rove: > ee margin. lothes esign he figure at the right is part of a clothing design pattern. In the figure, n n, #, and #. 7 k is isosceles with base, and ml ind the measures of all the numbered angles in the figure. ee margin. I >. Name two congruent triangles and tell how you can prove them congruent. eal-orld onnection k O k; roof areers clothing designer must carefully measure angles &I and &O are right '. > and segments to create rove: > O. ee back rove: > a sewing pattern. of book. I. iven: > I, I >,. iven: ', ', O hallenge 3. easoning raw a quadrilateral with 6 and 6, and its diagonals and intersecting at. abel your diagram to indicate the 3b. se O (efl. parallel sides. rop.) and alt. int. ' to O ; O ; O ; O show k Ok a. ist all the pairs of congruent segments that you can find in your diagram. (), O and b. riting xplain how you know that the segments you listed are congruent. O (). k Ok () roof Name a pair of overlapping congruent triangles in each diagram. and k Ok tate whether the triangles are congruent by,,,, or. (). hen O lan and write a proof. and O (). 4. iven: 5. iven: >, ', & > & 4 5. ', ee margin. > O 4. ssess & eteach oweroint esson uiz. Identify any common sides and angles in and. or xercises and 3, name a pair of congruent overlapping triangles. tate the theorem or postulate that proves them congruent.. k Ok; 3. I ki OkI; 4. lan a proof. iven:, rove: O by if k Ok. his congruence holds by if k Ok. how k Ok by. lesson quiz, chool.com, eb ode: aua O and l O l by ef. of # bisector. O so k O k by. l O l by. bisects l so l Ol and l and l are esson 4-7 sing orresponding arts of ongruent riangles 45 both rt. '. o l O l since they are compl. of O '. k O k by so O by. 9. ml 56; ml 56; ml3 34; ml4 90; ml5 ; ml6 34; ml7 34; ml8 68; ml9 4. k O k by ; O, l O l (iven) l O l (eflective rop. of O) k O k () 5. k O k by ;,, O (iven) l and l are rt. ' (ef. of ) (eflective rop. of O) k O k () 45

6 lternative ssessment sing the diagram below, have partners work together to find all the pairs of congruent triangles. or each pair, they should write a paragraph proof that the triangles are congruent. est rep esources or additional practice with a variety of test item formats: tandardized est rep, p. 53 est-aking trategies, p. 48 est-aking trategies with ransparencies est rep ultiple hoice hort esponse xtended esponse se the diagram at the right for xercises If m& = 5, what is m&? If m& = 30 and x = 7.4, what is the perimeter of #? If m& = 47, what is m&? he pentagon at the right is equilateral and equiangular. a. hat two triangles must be congruent to prove >? b. rite a proof to show >. a b. ee margin. 30. a. In the figure at the right, why is # > #? a e. ee margin. b. opy the figure. ark each angle that has measure x. c. hat is the value of x? xplain how you found your answer. d. hat is m&? e. hat is? xplain your answer. 3 m 3 m x x x ixed eview [] a. k O k b. O by if k O k by. ince k O k by, then l O l by and l O l because vertical ' are O. [] one part correct 30. [4] a. b. 3 m 3 m x x x x 46 c. x 30. In k ml + ml + ml O for elp n esson 4-6 p esson 3-8 esson qs. may vary, depending on pt. chosen. 46 hapter 4 ongruent riangles ubstituting, 90 + x + x + x 80. olving, x 30. d. 0; it is suppl. to a 60 l. e. 6 m; () 3. omplete the plan for a proof. iven: & and & are right angles, >. rove: # > # right lan: # and # are a. 9triangles with legs that are given to be b. 9. he hypotenuse is O congruent to itself by the c. 9 roperty of ongruence. eflexive # > # by the d. 9 heorem. onstructions raw a line p and a point not on p. onstruct the described line. 3. line n through so that n ' p 33. line r through so that r 6 p ee left. ee margin. rite an equation in point-slope form of the line that contains the given point and has the given slope. 34. (, -6); slope y ± 6 (x ) 35. (0, 5); slope y 5 (x 0) 36. (-3, 6); slope (0, 0); slope 3 y 0 3(x 0) y 6 (x ± 3) rite an equation in point-slope form of the line that contains the given points. 38. (, 4), (0, ) 39. (3, -5), (6, 0) 40. (-4, -3), (, -8) y 4 (x ) 5 y ± 5 3(x 3) 5 y ± 3 6(x ± 4) [3] 4 parts answered correctly [] 3 parts answered correctly [] parts answered correctly 33. r p