Triangle Congruences Day 2

Warm Up 1 9.8.2014 Find the missing angles below 2. Use Pythagorean Theorem to find the missing side y 70 x 70 40 40 What can we say about each pair of triangles? Congruent. THIS IS IMPORTANT TO DO BECAUSE WE’LL USE THIS PROCESS TO SKIP AAS AND HL. d 4 3

Essential Question #9 How can we prove triangles are congruent?

Today’s Objective Use triangle congruence postulates and theorems to prove that triangles are congruent.

Proving Triangles Congruent By the definition of Congruence, what do we need to show to prove two triangles are congruent? All corresponding angles are congruent All corresponding sides are congruent That’s six pairs! With Triangles, we have postulates that allow us to only need to show three pairs, but they have to be in a specific order

If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

If two sides and the included (between) angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Non-example of SAS: Why can’t we use SAS to show these triangles are congruent?

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.

We now have the following: SSS – side, side, side SAS – Side, Angle (between), Side ASA – Angle, Side (between), Angle

Examples Which Theorem proves the Triangles are congruent? 1.

Sometimes we have to mark Assumptions! Assumption #1: Reflexive Property If two triangles share a side, that side is congruent to itself

What do we know about the third angle if two are congruent? 2. JUST LIKE IN WARMUP, IF WE KNOW TWO ANGLES CONGRUENT, THEN THIRD IS TWO BECAUSE OF ANGLE SUM

Assumption #2: Third Angle If two pairs of corresponding angles are congruent, then the third pair is also congruent! Why? Triangle Angle Sum Theorem says the measures of the angles have to sum to 180!

What do we know about vertical angles? 3.

Assumption #3: Vertical Angles Vertical Angles are Congruent!

4.

Assumption #4: Third Side of a RIGHT Triangle If two pairs of corresponding sides are congruent in a RIGHT TRIANGLE, then the third pair is also congruent! Why? Pythagorean Theorem states that a2+b2=c2. This can only be true if a, b, and c are the same in both triangles

Additional Examples 5.

Additional Examples 6. JUST LIKE WITH WARM UP, IN RIGHT TRIANGLE IF TWO SIDES ARE SAME, THEN THIRD IS BECAUSE PYTH THM

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