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Proving Triangles Congruent

Proving Triangles Congruent. Triangle Congruency Short-Cuts. If you can prove one of the following short cuts, you have two congruent triangles SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-angle-side) HL (hypotenuse-leg) right triangles only!.

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Proving Triangles Congruent

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  1. Proving Triangles Congruent

  2. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles • SSS (side-side-side) • SAS (side-angle-side) • ASA (angle-side-angle) • AAS (angle-angle-side) • HL (hypotenuse-leg) right triangles only!

  3. Built – In Information in Triangles

  4. Identify the ‘built-in’ part

  5. Shared side SSS Vertical angles SAS Parallel lines -> AIA Shared side SAS

  6. SOME REASONS For Indirect Information • Def of midpoint • Def of a bisector • Vert angles are congruent • Def of perpendicular bisector • Reflexive property (shared side) • Parallel lines ….. alt int angles • Property of Perpendicular Lines

  7. This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

  8. HL( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. ASA HL

  9. Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

  10. Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:

  11. G K I H J 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. Ex 4 ΔGIH ΔJIK by AAS

  12. B A C D E 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. Ex 5 ΔABC ΔEDC by ASA

  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. Ex 6 E A C B D ΔACB ΔECD by SAS

  14. 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. Ex 7 J K L M ΔJMK ΔLKM by SAS or ASA

  15. 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. J T Ex 8 L K V U Not possible

  16. Problem #4 AAS Given Vertical Angles Thm Given AAS Postulate

  17. Problem #5 HL Given ABC, ADC right s, Prove: Given 1. ABC, ADC right s Given Reflexive Property HL Postulate

  18. 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?

  19. segments angles segments angles angles Given implies Congruent Parts midpoint parallel segment bisector angle bisector perpendicular

  20. Example Problem

  21. Step 1: Mark the Given … and what it implies

  22. Step 2: Mark . . . • Reflexive Sides • Vertical Angles … if they exist.

  23. Step 3: Choose a Method SSS SAS ASA AAS HL

  24. STATEMENTS REASONS S A S Step 4: List the Parts … in the order of the Method

  25. STATEMENTS REASONS Step 5: Fill in the Reasons S A S (Why did you mark those parts?)

  26. STATEMENTS REASONS 1. 2. 3. 4. 5. 1. 2. 3. 4. 5. Step 6: Is there more? S A S

  27. 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?

  28. 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.

  29. 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 meanssides AD and BC are congruent by CPCTC.

  30. Corresponding Parts of Congruent Triangles • A two column proof that sides AD and BC are congruent in the figure at right is shown below:

  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:

  32. 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

  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

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