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NAME ______________________________________________ DATE 4-5 ____________ PERIOD _____ Study Guide and Intervention Proving Congruence—ASA, AAS ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two triangles are congruent. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ASA Postulate Example Find the missing congruent parts so that the triangles can be proved congruent by the ASA Postulate. Then write the triangle congruence. a. B E A C D F Two pairs of corresponding angles are congruent, A D and C F. If the included sides A C and D F are congruent, then ABC DEF by the ASA Postulate. b. S X R T W Y R Y and S R X Y . If S X, then RST YXW by the ASA Postulate. Exercises What corresponding parts must be congruent in order to prove that the triangles are congruent by the ASA Postulate? Write the triangle congruence statement. 1. 2. C D B E W A Z 5. B 6. V R B Y A 4. A 3. X E D B C D T U C A C E Lesson 4-5 D S © Glencoe/McGraw-Hill 207 Glencoe Geometry NAME ______________________________________________ DATE 4-5 ____________ PERIOD _____ Study Guide and Intervention (continued) Proving Congruence—ASA, AAS AAS Theorem Another way to show that two triangles are congruent is the AngleAngle-Side (AAS) Theorem. AAS Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. You now have five ways to show that two triangles are congruent. • definition of triangle congruence • ASA Postulate • SSS Postulate • AAS Theorem • SAS Postulate Example 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 Postulate? C A A C by the Reflexive Property of congruence. The congruent angles cannot be 1 and 2, because A C would be the included side. If B D, then ABC ADC by the AAS Theorem. B A 1 2 C D Exercises In Exercises 1 and 2, draw and label ABC and DEF. Indicate which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem. 1. A D; B E 2. BC EF; A D B C F E A C D B F A 3. Write a flow proof. Given: S U; T R bisects STU. Prove: SRT URT TR bisects STU. Given E D S R T U STR UTR Def.of bisector S U SRT URT SRT URT Given AAS CPCTC RT RT Refl. Prop. of © Glencoe/McGraw-Hill 208 Glencoe Geometry NAME ______________________________________________ DATE 4-5 ____________ PERIOD _____ Skills Practice Proving Congruence—ASA, AAS Write a flow proof. 1. Given: N L J K M K Prove: JKN MKL J L K N M N L Given JK MK JKN MKL Given AAS JKN MKL Vertical are . 2. Given: AB C B A C D B bisects ABC. Prove: A D C D B D A C AB CB Given A C ABD CBD AD CD Given ASA CPCTC DB bisects ABC. ABD CBD Def. of bisector Given 3. Write a paragraph proof. F G Lesson 4-5 Given: DE || F G E G Prove: DFG FDE E D © Glencoe/McGraw-Hill 209 Glencoe Geometry NAME ______________________________________________ DATE 4-5 ____________ PERIOD _____ Practice Proving Congruence—ASA, AAS 1. Write a flow proof. Given: S is the midpoint of Q T . R Q || T U Prove: QSR TSU S is the midpoint of QT. Given QR || TU Given R T S Q U QS TS Def.of midpoint Q T Alt. Int. are . QSR TSU ASA QSR TSU Vertical are . 2. Write a paragraph proof. Given: D F G E bisects DEF. Prove: D G F G D G E F ARCHITECTURE For Exercises 3 and 4, use the following information. An architect used the window design in the diagram when remodeling an art studio. A B and C B each measure 3 feet. B A D C 3. Suppose D is the midpoint of A C . Determine whether ABD CBD. Justify your answer. 4. Suppose A C. Determine whether ABD CBD. Justify your answer. © Glencoe/McGraw-Hill 210 Glencoe Geometry NAME ______________________________________________ DATE 4-6 ____________ PERIOD _____ Study Guide and Intervention Properties of Isosceles Triangles An isosceles triangle has two congruent sides. The angle formed by these sides is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles. A • If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem) • If two angles of a triangle are congruent, then the sides opposite those angles are congruent. B C If A B C B , then A C. If A C, then A B C B . Example 1 Example 2 Find x. Find x. S C (4x 5) A (5x 10) B 3x 13 R BC BA, so mA mC. 5x 10 4x 5 x 10 5 x 15 T 2x mS mT, so SR TR. 3x 13 2x 3x 2x 13 x 13 Isos. Triangle Theorem Substitution Subtract 4x from each side. Add 10 to each side. Converse of Isos. Thm. Substitution Add 13 to each side. Subtract 2x from each side. Exercises Find x. 1. R P 40 2x 2. S 2x 6 T 3x 6 3. W V Q 4. D P K T (6x 6) 2x Q 5. G Y 3x 6. B Z T 30 3x 3x D 7. Write a two-column proof. Given: 1 2 Prove: A B C B L R x S B A 1 3 C D 2 E Statements © Glencoe/McGraw-Hill Reasons 213 Glencoe Geometry Lesson 4-6 Isosceles Triangles NAME ______________________________________________ DATE 4-6 ____________ PERIOD _____ Study Guide and Intervention (continued) Isosceles Triangles Properties of Equilateral Triangles An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles. 1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measures 60°. Example Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle. A P 1 Proof: 2 Q B C Statements Reasons Q || B C . 1. ABC is equilateral; P 2. mA mB mC 60 3. 1 B, 2 C 4. m1 60, m2 60 5. APQ is equilateral. 1. Given 2. Each of an equilateral measures 60°. 3. If || lines, then corres. s are . 4. Substitution 5. If a is equiangular, then it is equilateral. Exercises Find x. 1. 2. D 6x 5 F 6x 4. 4x V 5. Q 40 60 L Y 4x 4 Glencoe/McGraw-Hill 4x 60 H O A D 1 2 B C Proof: © R M 7. Write a two-column proof. Given: ABC is equilateral; 1 2. Prove: ADB CDB Statements KLM is equilateral. 6. X Z K M H 3x 8 60 R 3x N 5x J E P 3. L G Reasons 214 Glencoe Geometry NAME ______________________________________________ DATE 4-6 ____________ PERIOD _____ Skills Practice Isosceles Triangles Refer to the figure. Lesson 4-6 C 1. If A C A D , name two congruent angles. B D 2. If B E B C , name two congruent angles. E A 3. If EBA EAB, name two congruent segments. 4. If CED CDE, name two congruent segments. ABF is isosceles, CDF is equilateral, and mAFD 150. Find each measure. 5. mCFD 6. mAFB 7. mABF 8. mA A E F B L 9. If mRLP 100, find mBRL. 10. If mLPR 34, find mB. R 11. Write a two-column proof. P D E Given: CD C G E D G F Prove: C E C F Glencoe/McGraw-Hill D 35 In the figure, P L R L and L R B R . © C B C F G 215 Glencoe Geometry NAME ______________________________________________ DATE 4-6 ____________ PERIOD _____ Practice Isosceles Triangles Refer to the figure. R 1. If R V R T , name two congruent angles. S V 2. If R S S V , name two congruent angles. T U 3. If SRT STR, name two congruent segments. 4. If STV SVT, name two congruent segments. Triangles GHM and HJM are isosceles, with G H M H and H J M J . Triangle KLM is equilateral, and mHMK 50. Find each measure. J K L M H 5. mKML 6. mHMG 7. mGHM G 8. If mHJM 145, find mMHJ. 9. If mG 67, find mGHM. 10. Write a two-column proof. Given: DE || B C 1 2 Prove: A B A C E 2 3 C A 1 D 4 B 11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle. © Glencoe/McGraw-Hill 216 n col Lin ks Haw Glencoe Geometry