Proving Triangle Congruence Proof in mathematicss.ppt
- 2. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles 1.SAS (side-angle-side) If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
- 3. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles 2. ASA (angle-side-angle) If two angles and the included side of one triangle are congruent to the corresponding two angles and an included side of another triangle, then the triangles are congruent.
- 4. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles 3. SSS (side-side-side) If three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent.
- 5. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles 4. AAS (angle-angle-side) If two angles and the non-included side of one triangle are congruent respectively to the two angles and the non-included side of another triangle, then the two triangles are congruent.
- 6. Built – In Information in Triangles • Vertical Angle- a pair of opposite angles that form when two lines intersect. • Reflexive Property- shared angle and shared side of a two triangle.
- 7. Identify the ‘built-in’ part Identify the ‘built-in’ part
- 8. Shared side Shared side Shared side Shared side Vertical angles Vertical angles SAS SAS SAS SAS SSS SSS
- 9. SOME REASONS For Indirect SOME REASONS For Indirect Information Information • Def of midpoint Def of midpoint • Def of a bisector Def of a bisector • Vert angles are congruent Vert angles are congruent • Def of perpendicular bisector Def of perpendicular bisector • Reflexive property (shared side) Reflexive property (shared side) • Parallel lines Parallel lines • Property of Perpendicular Lines Property of Perpendicular Lines
- 10. This is called a common side. This is called a common side. It is a side for both triangles. It is a side for both triangles. We’ll use the reflexive property. We’ll use the reflexive property.
- 11. Name That Postulate Name That Postulate (when possible) SAS SAS SAS SAS SAS SAS Reflexive Property Vertical Angles Vertical Angles Reflexive Property SSA SSA
- 12. Let’s Practice Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: B D For AAS: A F AC FE
- 13. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. ΔGIH ΔJIK by AAS G I H J K Ex 4
- 14. ΔABC ΔEDC by ASA B A C E D Ex 5 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
- 15. ΔACB ΔECD by SAS B A C E D Ex 6 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
- 16. ΔJMK ΔLKM by SAS or ASA J K L M Ex 7 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
- 17. Problem #4 Statements Reasons AAS Given Given Vertical Angles Thm AAS Postulate Given: A C BE BD Prove: ABE CBD E C D A B 4. ABE CBD 38
- 18. Problem #5 3. AC AC Statements Reasons C B D AHL Given Given Reflexive Property HL Postulate 4. ABC ADC 1. ABC, ADC right s AB AD Given ABC, ADC right s, Prove: AB AD ABC ADC 39
- 19. Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles / Vertical Angles Also: mark info implied by given info. 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more? 40
- 20. Given implies Congruent Parts midpoint parallel segment bisector angle bisector perpendicular segments angles segments angles angles 41
- 21. Example Problem C B D A Given: AC bisects BAD AB AD Prove: ABC ADC 42
- 22. Step 1: Mark the Given … and what it implies C B D A Given: AC bisects BAD AB AD Prove: ABC ADC 43
- 23. •Reflexive Sides •Vertical Angles Step 2: Mark . . . … if they exist. C B D A Given: AC bisects BAD AB AD Prove: ABC ADC 44
- 24. Step 3: Choose a Method SSS SAS ASA AAS HL C B D A Given: AC bisects BAD AB AD Prove: ABC ADC 45
- 25. Step 4: List the Parts STATEMENTS REASONS … in the order of the Method C B D A Given: AC bisects BAD AB AD Prove: ABC ADC BAC DAC AB AD AC AC S A S 46
- 26. Step 5: Fill in the Reasons (Why did you mark those parts?) STATEMENTS REASONS C B D A Given: AC bisects BAD AB AD Prove: ABC ADC BAC DAC AB AD AC AC Given Def. of Bisector Reflexive (prop.) S A S 47
- 27. S A S Step 6: Is there more? STATEMENTS REASONS C B D A Given: AC bisects BAD AB AD Prove: ABC ADC BAC DAC AB AD AC AC Given AC bisects BAD Given Def. of Bisector Reflexive (prop.) ABC ADC SAS (pos.) 1. 2. 3. 4. 5. 1. 2. 3. 4. 5. 48
- 28. Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more? 54
- 29. Using CPCTC in Proofs • According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. • This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. • This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent. 55
- 30. Corresponding Parts of Congruent Triangles • For example, can you prove that sides AD and BC are congruent in the figure at right? • The sides will be congruent if triangle ADM is congruent to triangle BCM. – Angles A and B are congruent because they are marked. – Sides MA and MB are congruent because they are marked. – Angles 1 and 2 are congruent because they are vertical angles. – So triangle ADM is congruent to triangle BCM by ASA. • This means sides AD and BC are congruent by CPCTC. 56
- 31. Corresponding Parts of Congruent Triangles • A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Given Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC 57
- 32. Corresponding Parts of Congruent Triangles • A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Given Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC 58
- 33. Corresponding Parts of Congruent Triangles • Sometimes it is necessary to add an auxiliary line in order to complete a proof • For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU Given UF @ UF reflexive prop. DFRU @ DFOU SSS ÐR @ ÐO CPCTC 59
- 34. Corresponding Parts of Congruent Triangles • Sometimes it is necessary to add an auxiliary line in order to complete a proof • For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU Given UF @ UF Same segment DFRU @ DFOU SSS ÐR @ ÐO CPCTC 60