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4.5 ASA and AAS2 ink.notebook November 09, 2016 Page 161 Page 162 Page 160 4.5 ASA and AAS Page 163 Page 164 Page 165 1 4.5 ASA and AAS2 ink.notebook Lesson Objectives Standards November 09, 2016 Lesson Notes Lesson Objectives Standards Lesson Notes 4.5 ASA and AAS After this lesson, you should be able to successfully use ASA and AAS to prove triangles are congruent. Press the tabs to view details. Press the tabs to view details. ANGLESIDEANGLE (ASA) Lesson Objectives Standards Lesson Notes G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 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. and Angle ∠C ≅ ______, G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. E B G.CO.10 Prove theorems about triangles. A C D F 2 4.5 ASA and AAS2 ink.notebook November 09, 2016 For each diagram, determine which pairs of triangles can be proved congruent by the ASA Postulate. b) H a) G E F D 1. Given: AB Ç CD, ÚCBD ¤ ÚADB Prove: ÆABD ¤ ÆADB A D Statements B C Reasons 1. AB Ç CD 2. ÚCBD ¤ ÚADB 1. 3. ÚABD ¤ ÚBDC 3. 4. BD = BD 4. 5. ÆABD ¤ ÆADB 5. 2. 3 4.5 ASA and AAS2 ink.notebook November 09, 2016 2. Given: ÚS ¤ ÚV and T is the midpoint of SV Prove: ÆRTS ¤ ÆUTV R U S Statements V T Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. ANGLEANGLESIDE (AAS) CONGRUENCE THEOREM If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. If Angle ∠A ≅ _________, Angle ∠C, ≅ ________, B E A D C F 4 4.5 ASA and AAS2 ink.notebook November 09, 2016 For each diagram, determine which pairs of triangles can be proved congruent by the AAS Postulate. c) d) B D A C 3. In the diagram, ∠BCA ¤ ∠DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem? B A 1 2 C D 5 4.5 ASA and AAS2 ink.notebook November 09, 2016 Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. 4. 6. 5. Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence. 7. A C T 9. 8. M N O P V U B E D Q P R Q 6 4.5 ASA and AAS2 ink.notebook November 09, 2016 Flow Chart Proof: 10. Given: CF bisects ÚACE and ÚBFD Prove: ÆCBF ¤ ÆCDF C 3 D 4 B 2 1 A CF bisects ÚACE ÚACF ¤ ÚECF F E CF bisects ÚBFD ÚBCF ¤ ÚDCF CF ¤ CF ÆCBF ¤ ÆCDF 11. Given: BC Ç EF, AB = ED, ÚC ¤ ÚF Prove: ÆABC ¤ ÆDEF Statements A Reasons B C D E F 7 4.5 ASA and AAS2 ink.notebook November 09, 2016 Term/Postulate Abbreviation Included Side Angle-Side-Angle Picture The side between two angles. It is in the middle of the angles. Included Angle The angle formed by two sides. It is in the middle of the two sides. Side-Side-Side If 3 sides of 2 è's are ¤, then the 2 è are ¤ Side-Angle-Side If 2 sides & the included Ú are ¤ in 2 è's, then the 2 è are ¤ Picture Definition/Explanation Term/Postulate Abbreviation Definition/Explanation If 2 Ú's and the included side are ¤ in 2 è's, then the 2 è are ¤ Angle-Angle-Side If 2 Ú's and the NON-included side are ¤ in 2 è's, then the 2 è are Parts of a Right Triangle ¤ Hypotenuse: Side opposite the right Ú Leg Hypotenuse Leg: Sides that form a right Ú Leg HypotenuseLeg Congruence Corresponding Parts of Congruent Triangles are Congruent If hypotenuse and a leg of one RIGHT è's, If 2 è are ¤ to the other RIGHT then the 2 rt è' s è are ¤ are ¤, then the corresponding parts are also ¤ 8 4.5 ASA and AAS2 ink.notebook November 09, 2016 State if the two triangles are congruent. If they are, state how you know. 1. 2. 5. 6. PRACTICE 3. 4. 9 4.5 ASA and AAS2 ink.notebook November 09, 2016 Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence. 8. 7. Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence. Q 9. K J W T L M P S R Y X 10 4.5 ASA and AAS2 ink.notebook November 09, 2016 11 4.5 ASA and AAS2 ink.notebook November 09, 2016 Page 3 12 4.5 ASA and AAS2 ink.notebook November 09, 2016 13 4.5 ASA and AAS2 ink.notebook November 09, 2016 14 4.5 ASA and AAS2 ink.notebook November 09, 2016 Answers: Answers page 1: 1. ASA 3. ASA 5. ASA 7. SSS, èKJM ¤ èKLM 9. ASA, èXYT ¤ èTWX Answers page 2: Answers page 3 & 4: 1. C 3. D 5. D 7. C 9. A 11. D 15