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4.1 – 4.3 Triangle Congruency Geometry SWBAT: 1. Recognize congruent triangles 2. Prove that triangles are congruent using the ASA Congruence Postulate, SSS Congruence Postulate, SAS Congruence Postulate and the AAS Congruence Theorem 3. Use congruence postulates and theorems in real-life problems. Congruent • Means that corresponding parts are congruent, • Matching sides and angles will be congruent B A C Y X Z Naming • ORDER MATTERS!!!! Example 1 R • If two triangles are congruent… – Name all congruent angles X S – Name all congruent sides T Y Z Reminder… • If two angles of one triangle are congruent to two angles of another triangle then the 3rd angles are congruent Keep in mind • You can flip, turn or slide congruent triangles and they will maintain congruency!! SSS • If sides of one triangle are congruent to sides of a second triangle then the two triangles are congruent. Example 1 • Given STU with S(0, 5), T(0,0), and U(-2, 0) and XYZ with X (4, 8), Y (4, 5), and Z (6, 3), determine if STU XYZ Included angles SAS • If two sides and the included angle of one triangle are congruent to two sides and included angle of another triangle then the two triangles are congruent. Example 2 • Write a proof – Given: X is the midpoint of BD X is the midpoint of AC D - Prove: DXC BXA A X C B Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the B included side of one triangle are congruent to two angles and the C included side of a second triangle, then the triangles are congruent. A E F D • If asked to show that a pair of corresponding parts of two triangles are congruent you often prove the triangles are congruent the by definition of congruent triangles the parts are congruent (CPCTC) Example 3 • Given: VR RS UT SU SR TU • Prove: VR TU Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem • If two angles and a B non-included side of one triangle are congruent to two angles and the corresponding non- C included side of a second triangle, then the triangles are congruent. A E F D Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF Prove: ∆ABC ∆DEF B A E C F D Ex. 1 Developing Proof A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG. H E G F J Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. N M Q P Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, MP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. N M Q P Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD BC Prove: ∆ABD ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC . Use the fact that AD ║EC to identify a pair of congruent angles. C A B D E C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem 5. ASA Congruence Theorem Note: • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D C and A E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.