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Teacher: Mr. Joseph Triangle Proofs Name:__________________________ Date: 11/16/16 Solve: 1) 2) Given: YQL GDP, write a congruence statement for corresponding angles. 3) Given F G, S N, L K, and the fact that the triangles are congruent, write the triangle congruence statement. 4) Given AT = UG, TR = UJ, and the fact that the triangles are congruent, write the triangle congruence statement. 5) Given T S, BT = WS, YT = SP, and the fact that the triangles are congruent, write the triangle congruence statement. 6) Given: YVP NSC, write a congruence statement for corresponding sides. Given T N, Q R, WT = NS, and the fact that the triangles are congruent, write the triangle congruence statement. Name the Postulate or Theorem Used to Prove the Triangles Congruent (if any): 7) Given: __ __ __ __ GA AP, BG PR, 8) GAP PAR Given: __ __ __ __ TA NA, BA MA 9) Given: 10) Given: __ __ WP GF 11) Given: 12) Given: __ __ __ __ AC OD, CT GD, C D Fill in the reasons in the proof: 13) 14) Given: P D, PGI DGI Prove: PIG DIG. Statement Reason P PGI __ __ JA RI, Prove: D DGI __ __ IG IG PIG Given: DIG J JAM Statement __ __ JA RI J R R RIM. Reason RMI and AMJ are vertical angles RMI AMJ JAM 15) 16) Given: RIM Given: __ __ __ __ CO PI , OW GI, Prove: SIT Statement AND. Reason Prove: COW Statement __ __ CO PI __ __ SI AN S A T D SIT O AND 18) Given: RFP LFP, Prove: PLF Statement RFP LFP RPF LPF __ __ PF PF PLF I PIG. Reason __ __ OW GI I COW 17) O RPF PRF. Reason LPF PIG Given: __ __ __ __ NM NT, AN EN Prove: MEN Statement __ __ NM NT __ __ AN EN PRF N N TAN. Reason MEN TAN Write a Proof: 19) 20) Given: SEAT is a square. U C Prove: USE Given: CTA. Prove: 21) 22) Given: SIT AND. SIT AND. Given: __ __ __ __ TA NA, BA MA Prove: BAT MAN. Prove: Fill in the reasons in the proof: 23) 24) Given: Given: __ __ __ __ __ __ IT ER, LT TR, IT // ER Prove: __ __ LI // ET . Statement __ __ IT ER R TER ETR L and ETR are corresponding interior angles __ __ LI // ET __ __ SI AN A LTI and R are corresponding L Reason SIT __ __ IT // ER LIT __ __ ST AD __ __ IT DN __ __ LT TR LTI Prove: A Statement AND S S. Reason 25) 26) Given: Given: __ __ BA MA, Prove: __ __ IT ND . Statement __ __ SI AN A I N __ __ IT ND M Prove: __ __ MN BT . Statement __ __ BA MA S SIT Reason B B AND M BAT and MAN are vertical angles BAT BAT __ __ MN BT MAN MAN Reason EasyWorksheet Step By Step Answers User Name: J. Joseph Form #112511235638 Practice Problems 1) Given: YQL GDP, write a congruence statement for corresponding angles. Don't forget, since the order of the congruence explains the order of the corresponding pieces, that we can substitute in G in for Y everywhere, D in for Q everywhere, and P in for L. All we have to do is list out the angles: Y G, Q D, L P 2) Given T N, Q R, WT = NS, and the fact that the triangles are congruent, write the triangle congruence statement. All we have to do is write the letters in the same order for the first two pieces. Since the remaining side gives us the last set of letters, we get: TQW NRS 3) Given F G, S N, L K, and the fact that the triangles are congruent, write the triangle congruence statement. All we have to do is write the letters in the same order: FSL GNK 4) Given AT = UG, TR = UJ, and the fact that the triangles are congruent, write the triangle congruence statement. First, we need to figure out which angles are the same. In this case, both statements have T and U in common, so those must be the same angle. All we have to do is write the first letters in order: T __ __ U __ __ Next, we'll use the fact that AT = UG to fill in the second spot: TA __ UG __ And finally, we can fill in the last piece of information: TAR UGJ 5) Given T S, BT = WS, YT = SP, and the fact that the triangles are congruent, write the triangle congruence statement. All we have to do is write the first letters in order: T __ __ Next, we'll use the fact that BT = WS to fill in the second spot: And finally, we can fill in the last piece of information: TBY SWP S __ __ TB __ SW __ 6) Given: YVP NSC, write a congruence statement for corresponding sides. Don't forget, since the order of the congruence explains the order of the corresponding pieces, that we can substitute in N in for Y everywhere, S in for V everywhere, and C in for P. All we have to do is list out the sides: __ __ __ __ __ __ YV NS, VP SC, PY CN 7) Given: __ __ __ __ GA AP, BG PR, GAP PAR Since GAP and PAR are a linear pair, they are supplementary. Because they are also congruent, they must each be a right angle! In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or HL (HL is only valid in the case of right triangles!) Since the angle does not touch both segments, we would use HL (Hypotenuse-Leg). And we can write the congruence as BAG RAP. 8) Given: __ __ __ __ TA NA, BA MA First, notice that BAT and MAN are vertical angles, so they are congruent! In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as BAT MAN. 9) Given: In this case, we have two angles congruent. That's not enough information. (Remember, we need a side!) So the triangles are not necessarily congruent. 10) Given: __ __ WP GF First, since l // m, we have two sets of alternate interior angles that are congruent: OPW OFG and OWP OGF And, because they are vertical angles, POW FOG In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. In this case, we know all three angles are congruent, so either theorem would work! And we can write the congruence as POW FOG. 11) Given: Watch out! In this case the only thing the triangles have congruent to each other is one side length! That's not enough information. So the triangles are not necessarily congruent. 12) Given: __ __ __ __ AC OD, CT GD, C D In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as CAT DOG. 13) Given: P D, PGI DGI Prove: PIG DIG. Statement P PGI Reason D DGI __ __ IG IG PIG DIG __ __ First, notice that IG IG by the Reflexive Property. In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show the two triangles are congruent. And we can write the congruence as PIG DIG. So we have the following proof when we are done: Statement P Reason Given D DGI Given PGI __ __ IG IG Reflexive Property DIG AAS PIG 14) Given: __ __ JA RI, J Prove: R JAM RIM. Statement Reason __ __ JA RI J R RMI and AMJ are vertical angles RMI AMJ JAM RIM First, notice that AMJ and RMI are vertical angles, so they are congruent! In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show the two triangles are congruent. And we can write the congruence as JAM RIM. So we have the following proof when we are done: Statement __ __ JA RI J Reason Given Given R RMI and AMJ are vertical angles Given RMI AMJ Vertical Angle Theorem JAM RIM AAS 15) Given: Prove: SIT Statement __ __ SI AN S A T D SIT AND. Reason AND In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show the two triangles are congruent. And we can write the congruence as SIT AND. So we have the following proof when we are done: Statement __ __ SI AN Reason Given S A Given T D Given AND AAS SIT 16) Given: __ __ __ __ CO PI , OW GI, Prove: COW Statement O I PIG. Reason __ __ CO PI __ __ OW GI O COW I PIG In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as COW PIG. So we have the following proof when we are done: Statement Reason __ __ CO PI Given __ __ OW GI Given O Given I PIG SAS COW 17) Given: RFP Prove: LFP, PLF Statement RFP LFP RPF LPF RPF PRF. LPF Reason __ __ PF PF PLF PRF First, notice that we must be using the smallest triangles -- since we are given sides and/or angles that don't appear in the larger triangles. __ First, notice that PF __ PF by the Reflexive Property. In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent. And we can write the congruence as PLF PRF. So we have the following proof when we are done: Statement RFP Reason LFP Given LPF Given RPF __ __ PF PF Reflexive Property PRF ASA PLF 18) Given: __ __ __ __ NM NT, AN EN Prove: MEN Statement __ __ NM NT TAN. Reason __ __ AN EN N N MEN TAN First, notice that we must be using the larger triangles -- since we are given sides and/or angles that don't appear in the smaller triangle. Also, N N by the Reflexive Property. In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as MEN TAN. So we have the following proof when we are done: Statement __ __ NM NT Reason Given __ __ AN EN N Given Reflexive Property N TAN SAS MEN 19) Given: SEAT is a square. U C Prove: USE CTA. First, since SEAT is a square, we know that ESU angles). ATC (they are both right __ __ And we know that SE AT. In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show the two triangles are congruent. And we can write the congruence as USE CTA. So we have the following proof when we are done: Statement U Given C SEAT is a square ESU and ESU 20) Given: Given ATC are right angles Definition of a Square ATC __ __ SE AT. USE Reason All Right Angles are Congruent Definition of a Square CTA AAS Prove: SIT AND. In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show the two triangles are congruent. And we can write the congruence as SIT AND. So we have the following proof when we are done: Statement __ __ SI AN Reason Given S A Given T D Given SIT AND AAS 21) Given: __ __ __ __ TA NA, BA MA Prove: BAT MAN. First, notice that BAT and MAN are vertical angles, so they are congruent! In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as BAT MAN. So we have the following proof when we are done: Statement Reason __ __ TA NA Given __ __ BA MA Given BAT and MAN are vertical angles Given BAT MAN Vertical Angle Theorem BAT MAN SAS 22) Given: Prove: SIT AND. In this case, all sides of one triangle are congruent to all sides of the other triangle, so we use SSS (Side-Side-Side) to show the two triangles are congruent. And we can write the congruence as SIT AND. So we have the following proof when we are done: Statement __ __ ST AD Reason Given __ __ SI AN Given __ __ IT DN Given SIT AND SSS 23) Given: __ __ __ __ __ __ IT ER, LT TR, IT // ER Prove: __ __ LI // ET . Statement Reason __ __ IT ER __ __ LT TR __ __ IT // ER LTI and R are corresponding LTI R LIT TER L ETR L and ETR are corresponding interior angles __ __ LI // ET __ __ First, notice we have IT //ER, so we have corresponding angles: So LTI R. In this case, we have two sides congruent and one angle. This means we are either going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.) Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to show the two triangles are congruent. And we can write the congruence as LIT TER. So we have the following proof when we are done: Statement Reason __ __ IT ER Given __ __ LT TR Given __ __ IT // ER Given LTI and R are corresponding Given LTI R Corresponding Angles Theorem LIT TER SAS L L and CPCTC ETR ETR are corresponding interior angles Given __ __ LI // ET Corresponding Angle Converse 24) Given: Prove: A Statement __ __ ST AD __ __ SI AN __ __ IT DN SIT AND S. Reason A S In this case, all sides of one triangle are congruent to all sides of the other triangle, so we use SSS (Side-Side-Side) to show the two triangles are congruent. And we can write the congruence as SIT AND. So we have the following proof when we are done: Statement Reason __ __ ST AD Given __ __ SI AN Given __ __ IT DN Given AND SSS SIT A CPCTC S 25) Given: Prove: __ __ IT ND . Statement __ __ SI AN S A I N SIT AND Reason __ __ IT ND In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent. And we can write the congruence as SIT AND. So we have the following proof when we are done: Statement Reason __ __ SI AN Given S A Given I N Given AND ASA SIT __ __ IT ND CPCTC 26) Given: __ __ BA MA, B M Prove: __ __ MN BT . Statement __ __ BA MA B M BAT and BAT MAN are vertical angles MAN Reason BAT MAN __ __ MN BT First, notice that BAT and MAN are vertical angles, so they are congruent! In this case, we have two angles congruent and one side. This means we are either going to use AAS or ASA. Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent. And we can write the congruence as BAT MAN. So we have the following proof when we are done: Statement __ __ BA MA Reason Given M Given BAT and MAN are vertical angles Given B BAT MAN Vertical Angle Theorem BAT MAN ASA __ __ MN BT CPCTC All rights reserved. This page is copyright 1998 Triple Threat Inc. Any violators will be prosecuted through full extent of the law.