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4.5 Prove Triangles Congruent by ASA and AAS Goal Your Notes p Use two more methods to prove congruences. VOCABULARY Flow proof POSTULATE 21: ANGLE-SIDE-ANGLE (ASA) CONGRUENCE POSTULATE If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. If Angle ∠A > } Side AC > Angle ∠C > then , B E , and , nABC > A C D F . THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS) CONGRUENCE THEOREM If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. If Angle ∠A > Angle ∠C > } Side BC > then 104 Lesson 4.5 • Geometry Notetaking Guide nABC > , B E , and , A C D F . Copyright © Holt McDougal. All rights reserved. 4.5 Prove Triangles Congruent by ASA and AAS Goal Your Notes p Use two more methods to prove congruences. VOCABULARY Flow proof A flow proof uses arrows to show the flow of a logical argument. POSTULATE 21: ANGLE-SIDE-ANGLE (ASA) CONGRUENCE POSTULATE If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. If Angle ∠A > ∠D , } } Side AC > DF , and Angle ∠C > ∠F , then B A E C D F nABC > nDEF . THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS) CONGRUENCE THEOREM If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. If Angle ∠A > ∠D , Angle ∠C > ∠F , and } } Side BC > EF , then 104 Lesson 4.5 • Geometry Notetaking Guide B A E C D F nABC > nDEF . Copyright © Holt McDougal. All rights reserved. Your Notes Example 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b. a. c. Solution a. There is not enough information to prove the triangles are known to be are congruent, because no congruent. pair of b. Two pairs of angles and a sides are congruent. The triangles are congruent by . the c. The vertical angles are congruent, so two pairs of angles and their are congruent. The triangles are congruent by the . Checkpoint Can nSTW and nVWT be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. 1. S V 60° 60° U T 2. S W T W Copyright © Holt McDougal. All rights reserved. V Lesson 4.5 • Geometry Notetaking Guide 105 Your Notes Example 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b. a. c. Solution a. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. b. Two pairs of angles and a non-included pair of sides are congruent. The triangles are congruent by the AAS Congruence Theorem . c. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate . Checkpoint Can nSTW and nVWT be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. 1. S V 60° 60° U T W Yes; AAS Congruence Theorem 2. S T W V Yes; ASA Congruence Postulate Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Geometry Notetaking Guide 105 Your Notes Write a flow proof Example 2 In the diagram, ∠1 > ∠4 and } CF bisects ∠ACE. Write a flow proof to show nCBF > nCDF. Solution } Given ∠1 > ∠4, CF bisects ∠ACE. C B 2 1 A 3 D 4 F E Prove nCBF > nCDF ∠1 > ∠4 . } CF bisects. . ∠ACE. ∠1 and ∠2 are ∠3 and ∠4 are Linear Pair Postulate ∠2 > } } CF > CF ∠ACF > Def. of ∠ bisector Congruent Supps. Thm. ∠CBF > ∠CDF Checkpoint Complete the following exercise. } 3. In Example 2, suppose it is given that CF bisects ∠ACE and ∠BFD. Write a flow proof to show nCBF > nCDF. 106 Lesson 4.5 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Write a flow proof Example 2 In the diagram, ∠1 > ∠4 and } CF bisects ∠ACE. Write a flow proof to show nCBF > nCDF. Solution } Given ∠1 > ∠4, CF bisects ∠ACE. C B 2 1 A 3 D 4 F E Prove nCBF > nCDF ∠1 > ∠4 ∠1 and ∠2 are supplements . } CF bisects. ∠3 and ∠4 are supplements . ∠ACE. Given Given Linear Pair Postulate } } CF > CF Reflexive Prop. ∠2 > ∠3 Congruent Supps. Thm. ∠ACF > ∠ECF Def. of ∠ bisector ∠CBF > ∠CDF AAS Congruence Theorem Checkpoint Complete the following exercise. } 3. In Example 2, suppose it is given that CF bisects ∠ACE and ∠BFD. Write a flow proof to show nCBF > nCDF. } } CF bisects ∠ACE. CF bisects ∠BFD. Given ∠ACF > ∠ECF Def. of ∠ bisector Given } } CF > CF Reflexive Prop. ∠BFC > ∠DFC Def. of ∠ bisector ∠CBF > ∠CDF ASA Congruence Postulate 106 Lesson 4.5 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Example 3 Choose a postulate or theorem Games You and a friend are trying to find a flag hidden in the woods. Your friend is standing 75 feet away from you. When facing each other, the angle from you to the flag is 728 and the angle from your friend to the flag is 538. Is there enough information to locate the flag? Solution The locations of you, your friend, and the flag form a triangle. The measures of and an of the triangle are known. 72° 75 ft 53° By the , all triangles with these measures are congruent. So, the triangle formed is unique and the flag location is given by the . Checkpoint Complete the following exercise. 4. Theater Two actors are standing apart from each other on the edge of a stage. Spotlights are located and pointed as shown in the diagram. Can one of the actors move to another location on the stage without changing any of the angles of the triangle, without changing the distance to the other actor, and without requiring a spotlight to move? 20 ft 40° Homework Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Geometry Notetaking Guide 107 Your Notes Example 3 Choose a postulate or theorem Games You and a friend are trying to find a flag hidden in the woods. Your friend is standing 75 feet away from you. When facing each other, the angle from you to the flag is 728 and the angle from your friend to the flag is 538. Is there enough information to locate the flag? Solution The locations of you, your friend, and the flag form a triangle. The measures of two angles and an included side of the triangle are known. 72° 75 ft 53° By the ASA Congruence Postulate , all triangles with these measures are congruent. So, the triangle formed is unique and the flag location is given by the third vertex . Checkpoint Complete the following exercise. 4. Theater Two actors are standing apart from each other on the edge of a stage. Spotlights are located and pointed as shown in the diagram. Can one of the actors move to another location on the stage without changing any of the angles of the triangle, without changing the distance to the other actor, and without requiring a spotlight to move? 20 ft 40° Homework No. The measures of two angles and a nonincluded side of the triangle are known. By the AAS Congruence Theorem, all triangles with these measures are congruent. To create a different congruent triangle, one of the actors would have to move to a location that is off the stage, and the spotlight following that actor would have to move. Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Geometry Notetaking Guide 107