4-3. Triangle Congruence by ASA and AAS. Content Standard. Essential Understanding You can prove that two triangles are congruent

Transcription

1 4-3 riangle ongruence by and ontent tandard G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. bjective o prove two triangles congruent using the ostulate and the heorem h no! he school s photocopier is not working correctly. he copies all have some ink missing. elow are two photocopies of the same geometry worksheet. hich triangles are congruent? How do you know? Use what you already know about proving triangles congruent. hat is your plan for finding an answer? MHMIL I You already know that triangles are congruent if two pairs of sides and the included angles are congruent (). You can also prove triangles congruent using other groupings of angles and sides. ssential Understanding You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove triangles congruent by using one pair of corresponding sides and two pairs of corresponding angles. ostulate 4-3 ngle-ide-ngle () ostulate ostulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. If...,, hen hapter 4 ongruent riangles

2 roblem 1 Using hich two triangles are congruent by? xplain. rom the diagram you know U V UV U V o use, you need two pairs of congruent angles and a pair of included congruent sides. You already have pairs of congruent angles. o, identify the included side for each triangle and see whether it has a congruence marking. In UV, UV is included between U and V and has a congruence marking. In, is included between and and has a congruence marking. In, is included between and but does not have a congruence marking. ince U, UV, and V, UV. Got It? 1. hich two triangles are congruent by? xplain. G H I roblem 2 riting a Using ecreation Members of a teen organization are building a miniature golf course at your town s youth center. he design plan calls for the first hole to have two congruent triangular bumpers. rove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. an you use a plan similar to the plan in roblem 1? Yes. Use the diagram to identify the included side for the marked angles in each triangle. Given:,, and are right angles rove: : because all right angles are congruent, and you are given that. and are included sides between the two pairs of congruent angles. You are given that. hus, by. Lesson 4-3 riangle ongruence by and 235

3 Got It? 2. Given:,, and are right angles rove: You can also prove triangles congruent by using two angles and a nonincluded side, as stated in the theorem below. heorem 4-2 ngle-ngle-ide () heorem heorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. If...,, hen... of heorem 4-2: ngle-ngle-ide heorem Given:,, rove: Given hird ngles heorem Given Given You have seen and used three methods of proof in this book two-column, paragraph, and flow proof. ach method is equally as valid as the others. Unless told otherwise, you can choose any of the three methods to write a proof. Just be sure your proof always presents logical reasoning with justification. 236 hapter 4 ongruent riangles

4 How does information about parallel sides help? You will need another pair of congruent angles to use. hink back to what you learned in hapter 3. is a transversal here. roblem 3 riting a Using M Given: /M > /K, M 6 K rove: nm > nk K tatements easons 1) /M > /K 1) Given 2) M 6 K 2) Given 3) /M > /K 3) If lines are 6, then alternate interior ' are >. 4) > 4) eflexive roperty of ongruence 5) nm > nk 5) Got It? 3. a. Given: / > /, bisects / rove: n > n b. easoning In roblem 3, how could you prove that nm > nk by? xplain. roblem 4 etermining hether riangles re ongruent Multiple hoice Use the diagram at the right. hich of the following statements best represents the answer and justification to the question, Is ki ku? Yes, the triangles are congruent by. an you eliminate any of the choices? Yes. If ni nu then & and &U would be corresponding angles. You can eliminate choice. o, and are not corresponding sides. Yes, the triangles are congruent by. o, / and /U are not corresponding angles. he diagram shows that two pairs of angles and one pair of sides are congruent. he third pair of angles is congruent by the hird ngles heorem. o prove these triangles congruent, you need to satisfy or. U I and both fail because and are not included between the same pair of congruent corresponding angles, so they are not corresponding sides. he triangles are not necessarily congruent. he correct answer is. Got It? 4. re n and ni congruent? xplain. I Lesson 4-3 riangle ongruence by and 237

5 Lesson heck o you know H? 1. In, which side is included between and? 2. In M, is included between which angles? hich postulate or theorem could you use to prove? 3. o you U? 5. ompare and ontrast How are the ostulate and the ostulate alike? How are they different? 6. rror nalysis Your friend asks you for help on a geometry exercise. elow is your friend s paper. hat error did your friend make? xplain. MHMIL I easoning uppose I and GI. hat else must you know in order to prove GHI by? y? ractice and roblem-olving xercises MHMIL I ractice ame two triangles that are congruent by. ee roblem X 9. H U V G I 10. eveloping omplete the paragraph proof by filling in the blanks. Given: LKM JKM, K LMK JMK rove: LKM JKM : LKM JKM and L M J LMK JMK are given. KM KM by the a. roperty of ongruence. o, LKM JKM by b.. ee roblem Given:, rove: 12. Given:, rove: 238 hapter 4 ongruent riangles

6 13. eveloping omplete the two-column proof by filling in the blanks. Given:, line bisects at rove: tatements easons 1) 1) Given 2) 2) a. 3) Line bisects at. 3) b. 4) c. 4) efinition of bisect 5) 5) d. ee roblem Given: V Y, 15. Given:,, Z bisects VY is the midpoint of rove: VZ YZ rove: V Z Y etermine whether the triangles must be congruent. If so, name the postulate or theorem that justifies your answer. If not, explain. ee roblem M U Z V Y pply 19. Given:, M 20. Given: JG HGJ, G JH rove: M rove: GJ HJG M G J H Lesson 4-3 riangle ongruence by and 239

7 21. hink bout a lan hile helping your family clean out the attic, you find the piece of paper shown at the right. he paper contains clues to locate a time capsule buried in your backyard. he maple tree is due east of the oak tree in your backyard. ill the clues always lead you to the correct spot? xplain. How can you use a diagram to help you? hat type of geometric figure do the paths and the marked line form? How does the position of the marked line relate to the positions of the angles? 22. onstructions Use a straightedge to draw a triangle. Label it JKL. onstruct M JKL so that the triangles are congruent by. 23. easoning an you prove that the triangles at the right are congruent? Justify your answer. 24. riting nita says that you can rewrite any proof that uses the heorem as a proof that uses the ostulate. o you agree with nita? xplain. 25. Given:,, 26. Given: 1 2, and H bisects. rove: rove: H H 1 2 H 27. raw a iagram raw two noncongruent triangles that have two pairs of congruent angles and one pair of congruent sides. 28. Given:, rove: hallenge 29. Given and, name as many pairs of congruent triangles as you can. 30. onstructions In at the right, 5, 9, and m 30. how that there is no congruence rule by constructing UV with UV, U, and m m, but with UV hapter 4 ongruent riangles

8 31. robability elow are six statements about the triangles at the right. X Y Z XY XZ YZ here are 20 ways to choose a group of three statements from these six. hat is the probability that three statements chosen at random from the six will guarantee that the triangles are congruent? X Y Z tandardized est rep / 32. uppose and. hat additional information do you need to prove that J by? J J 33. You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. ach photo is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest number of photos you can fit on your poster? hich of the following figures is a concave polygon? hort esponse 35. rite the converse of the true conditional statement below. hen determine whether the converse is true or false. If you are less than 18 years old, then you are too young to vote in the United tates. Mixed eview ould you use or to prove the triangles congruent? xplain. 36. L 37. ee Lesson 4-2. M Get eady! o prepare for Lesson 4-4, do xercises 38 and 39. or I LK, list the indicated parts. 38. congruent corresponding angles 39. congruent corresponding sides ee Lesson 4-1. Lesson 4-3 riangle ongruence by and 241