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# Download Ways to prove Triangles Congruant

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Ways to prove Triangles Congruent (SSS), (SAS), (ASA) 4-2 to 4-4 EXAMPLE 4 Find m Use the Third Angles Theorem BDC. SOLUTION A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, m ACD = 180° – 45° – 30° = 105° . ANSWER So, m ACD = m BDC = 105° by the definition of congruent angles. EXAMPLE 5 Prove that triangles are congruent Write a proof. GIVEN AD CB, DC ACD PROVE AB CAB, ACD CAD ACB CAB Plan for Proof a. Use the Reflexive Property to show that AC AC. b. Use the Third Angles Theorem to show that B D EXAMPLE 5 Prove that triangles are congruent Plan in Action STATEMENTS REASONS 1. AD CB, DC a. 2. AC AC. BA 1. Given 2. Reflexive Property of Congruence 3. b. 4. 5. ACD CAD B ACD CAB, ACB D 3. Given 4. Third Angles Theorem CAB 5. Definition of GUIDED PRACTICE for Examples 4 and 5 4. In the diagram, what is m DCN. SOLUTION CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75° GUIDED PRACTICE for Examples 4 and 5 5. By the definition of congruence, what additional information is needed to know that NDC NSR. SOLUTION Given : CN NR, DCN Proved : NDC CDN NRS NSR. NSR, (Proved from above sum) GUIDED PRACTICE for Examples 4 and 5 STATEMENT REASON CDN NSR Given DCN NRS Given Therefore DC RS, DN SN as angles are congruent their sides are congruent. EXAMPLE 1 Identify congruent parts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. SOLUTION The diagram indicates that Corresponding angles J Corresponding sides JK JKL T, K TS, KL TSR. S, SR, LJ L R RT EXAMPLE 2 Use properties of congruent figures In the diagram, DEFG SPQR. a. Find the value of x. b. Find the value of y. SOLUTION a. You know that FG FG = QR 12 = 2x – 4 16 = 2x 8=x QR. EXAMPLE 2 b. Use properties of congruent figures You know that F m F=m Q 68o = (6y + x)o 68 = 6y + 8 10 = y Q. EXAMPLE 3 Show that figures are congruent PAINTING If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain. SOLUTION From the diagram, A C and D B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC . EXAMPLE 3 Show that figures are congruent Then, 1 4 and 2 3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent. The diagram shows AJ CK , KD JB , and DA BC . By the Reflexive Property, JK KJ . All corresponding parts are congruent, so AJKD CKJB. GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH CDEF. 1. Identify all pairs of congruent corresponding parts. SOLUTION Corresponding sides: Corresponding angles: AB CD, BG DE, GH FE, HA FC A G C, B E, H D, F. GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH 2. Find the value of x and find m SOLUTION (a) You know that (4x+ 5)° = 105° 4x = 100 x = 25 (b) You know that H m H m F =105° H F F CDEF. H. GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH 3. Show that PTS CDEF. RTQ. SOLUTION In the given diagram PS QR, PT TR, ST TQ and Similarly all angles are to each other, therefore all of the corresponding points of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem. EXAMPLE 1 Use the SSS Congruence Postulate Write a proof. GIVEN PROVE Proof KL NL, KM KLM NM NLM It is given that KL NL and KM By the Reflexive Property, LM So, by the SSS Congruence Postulate, KLM NLM NM LN. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 1. DFG HJK SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Side DG HK, Side DF JH,and Side FG JK. So by the SSS Congruence postulate, Yes. The statement is true. DFG HJK. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 2. ACB CAD SOLUTION GIVEN : BC PROVE : PROOF: AD ACB CAD It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD. GUIDED PRACTICE for Example 1 Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 3. QPT RST SOLUTION GIVEN : QT TR , PQ SR, PT TS PROVE : RST PROOF: QPT It is given that QT TR, PQ SR, PT TS. So by SSS congruence postulate, QPT RST. Yes the statement is true EXAMPLE 1 Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD ABC PROVE CDA STATEMENTS S REASONS 1. BC DA 1. Given 2. BC AD 2. Given A 3. S 4. BCA AC DAC CA 3. Alternate Interior Angles Theorem 4. Reflexive Property of Congruence EXAMPLE 1 Use the SAS Congruence Postulate STATEMENTS 5. ABC CDA REASONS 5. SAS Congruence Postulate EXAMPLE 2 Use SAS and properties of shapes In the diagram, QS and RP pass through the center M of the circle. What can you conclude about MRS and MPQ? SOLUTION Because they are vertical angles, PMQ RMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MS are all equal. ANSWER MRS and MPQ are congruent by the SAS Congruence Postulate. for Examples 1 and 2 GUIDED PRACTICE In the diagram, ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides VU . of ABCD. Also, RT SU and SU 1. Prove that SVR UVR STATEMENTS REASONS 1. 1. Given 2. 3. 4. SV VU SVR RV RVU VR SVR UVR 2. Definition of line 3. Reflexive Property of Congruence 4. SAS Congruence Postulate GUIDED PRACTICE 2. Prove that for Examples 1 and 2 BSR DUT STATEMENTS REASONS 1. 1. Given 2. BS RBS 3. RS 4. DU BSR TDU 2. Definition of line 3. Given UT DUT 4. SAS Congruence Postulate EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem Write a proof. GIVEN PROVE WY XZ, WZ ZY, XY ZY WYZ XZY SOLUTION Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram. EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem STATEMENTS H 1. WY 4. ZY 2. Given 3. Definition of Z and Y are lines right angles WYZ and XZY are 4. Definition of a right triangle right triangles. L 5. ZY 6. 1. Given XZ 2. WZ ZY, XY 3. REASONS WYZ YZ 5. Reflexive Property of Congruence XZY 6. HL Congruence Theorem EXAMPLE 4 Choose a postulate or theorem Sign Making You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You know that RP QS and PQ PS . What postulate or theorem can you use to conclude that PQR PSR? EXAMPLE 4 Choose a postulate or theorem SOLUTION You are given that PQ PS . By the Reflexive Property, RP RP . By the definition of perpendicular lines, both RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent. ANSWER You can use the SAS Congruence Postulate to conclude PQR PSR that . GUIDED PRACTICE for Examples 3 and 4 Use the diagram at the right. 3. Redraw ACB and DBC side by side with corresponding parts in the same position. GUIDED PRACTICE for Examples 3 and 4 Use the diagram at the right. 4. Use the information in the diagram to ACB DBC prove that STATEMENTS H 1. AB 2. AC DC BC, DB BC B REASONS 1. Given 2. Given 3. Definition of lines 3. C 4. ACB and DBC are 4. Definition of a right triangle right triangles. GUIDED PRACTICE for Examples 3 and 4 STATEMENTS REASONS L 5. BC 5. Reflexive Property of Congruence 6. ACB CB DBC 6. HL Congruence Theorem