4.3 Prove Triangles Congruent

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1 4.3 Prove riangles Congruent b Goal p Use side lengths to prove triangles are congruent. Your Notes POULAE 9: IDE-IDE-IDE () CONGUENCE POULAE If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. B If ide } AB >, ide } BC >, and ide } CA >, then nabc >. A C Eample Use the Congruence Postulate rite a proof. Given } F > } H, G is the midpoint of } FH. Prove nfg > nhg F G Proof It is given that } F >. Point G is the midpoint of } FH, so. B the efleive Propert,. o, b the, nfg > nhg. H Checkpoint Decide whether the congruence statement is true. Eplain our reasoning.. nkl > nmkl 2. n > n L K M Copright Holt McDougal. All rights reserved. Lesson 4.3 Geometr Notetaking Guide 97

2 4.3 Prove riangles Congruent b Goal p Use side lengths to prove triangles are congruent. Your Notes POULAE 9: IDE-IDE-IDE () CONGUENCE POULAE If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. B If ide } AB > }, ide } BC > }, and ide } CA > }, then nabc > n. A C Eample Use the Congruence Postulate rite a proof. Given } F > } H, G is the midpoint of } FH. Prove nfg > nhg F G Proof It is given that } F > } H. Point G is the midpoint of } FH, so } FG > } HG. B the efleive Propert, } G > } G. o, b the Congruence Postulate, nfg > nhg. H Checkpoint Decide whether the congruence statement is true. Eplain our reasoning.. nkl > nmkl 2. n > n L K M rue; all corresponding sides are congruent. False; } À } Copright Holt McDougal. All rights reserved. Lesson 4.3 Geometr Notetaking Guide 97

3 Eample 2 Congruence in the coordinate plane Determine whether npq is congruent to the other triangles shown at the right. olution B counting, PQ 5 3 and Q 5 5. Use the distance formula to find P. d 5 Ï }}} ( 2 2 ) 2 ( 2 2 ) 2 P 5 Ï }}} 5 Ï } B the Congruence Postulate, an triangle with side lengths,, and will be congruent to npq. he distance from to is. he distance from to is. he distance from to is Ï }}} 5 Ï }. o, npq >. he distance from to is Ï }}} 5 Ï }. No side of npq has a length of Ï }, so npq n. P Q 3 Checkpoint Complete the following eercise. 3. ndfg has vertices D(22, 4), F(4, 4), and G(22, 2). nlmn has vertices L(23, 23), M(23, 3), and N(2, 23). Graph the triangles in the same coordinate plane and show that the are congruent. 98 Lesson 4.3 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

4 Eample 2 Congruence in the coordinate plane Determine whether npq is congruent to the other triangles shown at the right. olution 3 B counting, PQ 5 3 and Q 5 5. Use the distance formula to find P. d 5 Ï }}} ( 2 2 ) 2 ( 2 2 ) 2 P 5 Ï }}} (2 2 (23)) 2 (2 2 5) 2 5 Ï } 34 B the Congruence Postulate, an triangle with side lengths 3, 5, and Ï } 34 will be congruent to npq. he distance from to is 3. he distance from to is 5. he distance from to is Ï }}} (2 2 5) 2 (23 2 2) 2 5 Ï } 34. o, npq > n. he distance from to is Ï }}} (23 2 0) 2 (22 2 (23)) 2 5 Ï } 0. No side of npq has a length of Ï } 0, so npq À n. P Q Checkpoint Complete the following eercise. 3. ndfg has vertices D(22, 4), F(4, 4), and G(22, 2). nlmn has vertices L(23, 23), M(23, 3), and N(2, 23). Graph the triangles in the same coordinate plane and show that the are congruent. D M F G L N DG 5 LN 5 2, DF 5 LM 5 6, and FG 5 MN 5 Ï } 40, so ndfg > nlmn b the Congruence Postulate. 98 Lesson 4.3 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

5 Eample 3 olve a real-world problem tabilit Eplain wh the table with the diagonal legs is stable, while the one without the diagonal legs can collapse. olution he table with the diagonal legs forms triangles with side lengths. B the Congruence Postulate, these triangles, so the table is. he table without the diagonal legs is because there are man possible quadrilaterals with the given side lengths. Checkpoint Determine whether the figure is stable. Eplain our reasoning Homework Copright Holt McDougal. All rights reserved. Lesson 4.3 Geometr Notetaking Guide 99

6 Eample 3 olve a real-world problem tabilit Eplain wh the table with the diagonal legs is stable, while the one without the diagonal legs can collapse. olution he table with the diagonal legs forms triangles with fied side lengths. B the Congruence Postulate, these triangles cannot change shape, so the table is stable. he table without the diagonal legs is not stable because there are man possible quadrilaterals with the given side lengths. Checkpoint Determine whether the figure is stable. Eplain our reasoning Yes, the figure is stable. B the Congruence Postulate, the triangles formed cannot change shape, so it is stable. No, the figure is not stable. here are man possible quadrilaterals with the given side lengths. Homework Copright Holt McDougal. All rights reserved. Lesson 4.3 Geometr Notetaking Guide 99

8.7 Coordinate Proof with

8.7 Coordinate Proof with Quadrilaterals Goal Eample p Use coordinate geometr to prove properties of quadrilaterals. Determine if quadrilaterals are congruent Determine if the quadrilaterals with the given