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Warm-Up

Warm-Up. If m< J + m< E + m< R = 180°, then construct < R. 4.4 Prove Triangles Congruent by SAS and HL 4.5 Prove Triangles Congruent by ASA and AAS. Objectives: To discover and use shortcuts for showing that two triangles are congruent. Congruent Triangles (CPCTC).

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Warm-Up

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  1. Warm-Up If m<J + m<E + m<R = 180°, then construct <R.

  2. 4.4 Prove Triangles Congruent by SAS and HL4.5 Prove Triangles Congruent by ASA and AAS Objectives: • To discover and use shortcuts for showing that two triangles are congruent

  3. Congruent Triangles (CPCTC) Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent. • Corresponding sides are congruent • Corresponding angles are congruent

  4. Congruent Triangles Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

  5. Congruence Shortcuts? • Will one pair of congruent sides be sufficient? One pair of angles?

  6. Congruence Shortcuts? • Will two congruent parts be sufficient?

  7. Congruent Shortcuts? • Will three congruent parts be sufficient?

  8. Congruent Shortcuts? • Will three congruent parts be sufficient? Included Angle Included Side

  9. Congruent Shortcuts? • Will three congruent parts be sufficient?

  10. Investigation: Shortcuts Well, we know that SSS is a valid shortcut, and I’ll give you the hint that 2 others in the list do not work. We will test the remaining 5in class. For each of these, you will be given three pieces to form a triangle. If the shortcut works, one and only one triangle can be made with those parts. • Shortcuts?: • SSS • SSA • SAS • ASA • AAS • AAA √

  11. Copying an Angle • Put point of compass on B and pencil on C. Make a small arc.

  12. Congruence Shortcuts Side-Side-Side (SSS) Congruence Postulate: If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

  13. Congruence Shortcuts Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

  14. Congruence Shortcuts Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

  15. Congruence Shortcuts Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the two triangles are congruent.

  16. And One More! Hypotenuse-Leg (HL) Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

  17. Example 1 What is the length of the missing leg in the each of the right triangles shown? 12 Notice that the pieces given here correspond to SSA, which doesn’t work. Because of the Pythagorean Theorem, right triangles are an exception. Therefore, rt. triangles have theorems such as HL (hypotenuse-leg) and LL (leg-leg) 12

  18. Example 2 Determine whether the triangles are congruent in each pair. Yes, SAS No

  19. Example 3 Determine whether the triangles are congruent in each pair. Answer and explain which theorem in your notebook

  20. Example 4 Explain the difference between the ASA and AAS congruence shortcuts. Answer in your notebook.

  21. Example 5 TRY IT in your notebook! I will pick someone at random to work it on the board  Ain’t life GRAND!

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