Proving Congruent Triangles using Two Way Table
- 1. Congruent Triangles Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
- 2. The Triangle Congruence Postulates &Theorems LA HA LL HL FOR RIGHT TRIANGLES ONLY AAS ASA SAS SSS FOR ALL TRIANGLES
- 3. SSS postulate SSS (side, side, side) postulate If three sides of a triangle are congruent to its three corresponding sides of another triangle, then the two triangles are congruent.
- 4. AB ED , ≅ BC EF and ≅ CA FD ≅ ∆ABC ∆DEF ≅ Look at these two triangles
- 5. SAS postulate SAS Postulate (Side-Angle-Side) If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
- 6. Look at these triangles. AC ≅ XZ C ≅ Z CB ZY ≅ ∆ABC ≅ ∆XYZ
- 7. EXAMPLE 1 Write a proof. GIVEN PROVE STATEMENTS REASONS BC DA ≅ , BC AD ∆ABC ∆ ≅ CDA 1. Given 1. BC DA ≅ S Given 2. 2. BC AD 3. BCA ≅ DAC 3. Alternate Interior Angles Theorem A 4. 4. AC ≅ CA Reflexive property S
- 8. EXAMPLE 1 STATEMENTS REASONS 5. ABC ≅ CDA SAS Postulate 5.
- 9. Given: RS RQ and ST QT Prove: Δ QRT Δ SRT. Q R S T EXAMPLE 2
- 10. STATEMENT REASON ________ 1. RS RQ; ST QT 1. Given 2. RT RT 2. Reflexive 3. Δ QRT Δ SRT 3. SSS Postulate R Q S T EXAMPLE 2
- 11. ASA Postulate ASA Postulate (Angle-Side-Angle) If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
- 12. Look at these triangles. B ≅ E BC ≅ EF C ≅ F ∆ABC ∆ ≅ DEF
- 13. AAS Theorem AAS (Angle-Angle-Side) 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 triangles are congruent.
- 14. Look at these triangles. B ≅ E C ≅ F AC ≅ DF ∆ABC ∆ ≅ DEF
- 15. Given: AD║EC, BD BC Prove: ∆ABD ∆EBC EXAMPLE 4
- 16. Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given 3. If || lines, then alt. int. s are 4. Vertical Angles Theorem 5. ASA Congruence Postulate
- 17. EXAMPLE 5 GIVEN - EGF JGH, EF HJ PROVE - ∆EFG ∆JHG
- 18. EXAMPLE 5 STATEMENTS REASONS 1. EFG JHG 1. Given 2. EF HJ 2. Given 3. EGF JGH 3. Vertical angles theorem 4. ∆EFG ∆JHG 4. AAS Theorem
- 19. Given: YR MA and AR RM Prove: Δ MYR Δ AYR Y A R M Try to solve this.
- 20. CPCTC Theorem • CPCTC states that if two or more triangles are proven congruent by any method, then all of their corresponding angles and sides are congruent as well.
- 21. Given: YR MA and AR RM Prove: AY MY Y A R M Try to solve this.
- 22. To prove that triangles are congruent we are going to use these theorems and postulates. 1.The (SSS) Side-Side-Side postulate 2.The (SAS) Side-Angle-Side postulate 3.The (ASA) Angle-Side-Angle postulate 4.The (AAS) Angle-Angle-Side theorem
- 23. 2. GIVEN; DE CE, EA EB PROVE; ∆DAB ∆CBA 1. GIVEN; circle with center H AHB FHB PROVE; A F H A F B D E C B A Prove the following. ( 20 pts. )
- 24. Assignment 1.What is the HL theorem? 2.What is the LL theorem? • Reference; Plane Geometry for Secondary Schools