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Module 5 Lesson 2 – Part 1

Module 5 Lesson 2 – Part 1. Proving Triangles Congruent (Remember to print the Learning Guide notes that go with this lesson so you can use them as you follow along.). Triangles are congruent if they have the same shape and the same size.

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Module 5 Lesson 2 – Part 1

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  1. Module 5 Lesson 2 – Part 1 Proving Triangles Congruent (Remember to print the Learning Guide notes that go with this lesson so you can use them as you follow along.)

  2. Triangles are congruent if they have the same shape and the same size. Since triangles have 6 parts (3 angles and 3 sides), all 6 corresponding parts must be the same. For example,

  3. The order of the letters makes a HUGE difference so be careful! For these two triangles, we said because the letters matched up (<A <E, <B <F, <C <G) We could have also said since the letters still match up (just in a different order). But this statement would be incorrect because the letters do not match. (<C does NOT equal <F so they can’t be in the same spot.)

  4. Example Identify the congruent triangles in the picture (in other words, write a congruency statement). So one answer is Another CORRECT answer would be since the letters still match up correctly (just in a different order).

  5. Another example Find x if Step 1: Draw the two triangles and label everything. Step 2: Remember that <T = <F so you can fill in 5x + 45 for <T also. Step 3: Remember the 3 angles of a triangle always add up to 180. <R + <S + <T = 180 so 75 + 25 + 5x + 45 = 180 145 + 5x = 180 -145 -145 --------------------- 5x = 35 x = 7 is the answer! R D D R 75 75 25 25 5x+45 5x+45 S E E S T F T F 5x+45

  6. 5 shortcuts for proving triangles congruent • You don’t always have to prove all 3 angles and all 3 sides are equal. • You can use one of these shortcuts: • Side-Side-Side (SSS) • Side-Angle-Side (SAS) • Angle-Side-Angle (ASA) • Angle – Angle-Side (AAS) • Hypotenuse – Leg (HL) You MUST look at a picture to determine which one you can use! Let’s look at an example of each one.

  7. SSS – Side-Side-Side • This is the easiest one to spot!The three sides of one triangle are congruent to the three sides of the other. • The sides are either already marked like this: • OR you will be given information to use to mark them yourself.

  8. SAS – Side – Angle- Side • You will need 2 sides that are congruent to side in the other triangle AND you need 1 angle. • The angle MUST be right in between the two sides like this: Side 1 Side 2 The angle is right in between the two sides,where the two sides TOUCH!

  9. ASA – Angle – Side - Angle • You will need 2 ANGLES this time and only 1 side. • BUT, the side must be right in between the two angles. If you look at the side, there is one angle marked at EACH END of the side. This is ALWAYS what ASA looks like.

  10. AAS – Angle – Angle - Side • This is easily confused with ASA because there are 2 angles and 1 side marked. • For AAS, the side comes AFTER the two angles (it is NOT in between them). • AAS looks like this:

  11. HL – Hypotenuse Leg • HL only works with RIGHT triangles. • So you need 3 things: • A right angle box in each triangle • The hypotenuse of each triangle is marked the same • One of the legs is also marked the same NOTE: Some students think this looks like SSA but remember there is no such shortcut!

  12. All the sides and angles must be marked before you answer the question. • You are NOT allowed to make “extra” marks on the diagrams except for 3 situations. • The following slides show an example of each situation in which you are allowed to make extra marks.

  13. 1. You can mark Vertical angles.Example: This looks like only 2 sides are marked BUT you can mark the vertical angles in the middle. So the answer is SAS.

  14. 2. You can mark the side if the two triangles share it. (That is because the side is congruent to itself –the reflexive property.) Example: This looks like 1 side and 1 angle. BUT, they share the side in the middle so you can mark that. (We will use 2 tic marks since the other sides are already using 1 mark.) So the answer is really SAS.

  15. 3. You can mark angles if you have parallellines (angles like alternate interior). Remember with parallel lines, the angles that make a Z shape are alternate interior angles and they are equal. So for this problem, The arrows mean you have parallel lines. Connect them to make a Z or backwards Z shape. The angles inside the Z are congruent so mark them. You can also mark the side in the middle because they share it. ANSWER: SAS

  16. Review • There are only 5 ways to prove triangles congruent: • SSS • SAS • ASA • AAS • HL • If you can’t use one of these ways, then the triangles are NOT CONGRUENT! You are NOT allowed to use SSA or AAA. Those are not valid shortcuts!

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