LESSON TEN: CONGRUENCE CONUNDRUM

CONGRUENCE We have already discussed similarity in triangles. What things must two triangles have in order to be similar? – Identical Angles – Proportional Sides

CONGRUENCE If I have two angles that are identical to another triangle, what do I know about the third angle? The third angle will also be identical. It must be. 45 ⁰ 80 ⁰ 45 ⁰ 80 ⁰ ??

CONGRUENCE Therefore, the last item we will discuss is the AA Condition for similarity. If a triangle has two angles in common with another triangle, the triangles area at least similar. Why “at least”?

CONGRUENCE When two triangles have identical corresponding angles AND identical corresponding sides, we say they are congruent. While we do need to know the triangles are the same, there are several shortcuts we can use to prove it.

CONGRUENCE The first shortcut we’ll discuss is the Side Side Side Congruence Postulate or SSS for short. This simply says that if all corresponding sides are congruent, then the triangles are also congruent. Watch this!

CONGRUENCE For instance, given the figure below. We know that the middle segment equals itself, by the reflexive property. So because all corresponding sides are congruent, the two triangles are congruent.

CONGRUENCE The next shortcut we will discover is the Side Angle Side Postulate or SAS for short. This states that if two corresponding sides and their included angle are congruent, then the two triangles are congruent.

CONGRUENCE An included angle is one that is made by the two sides in question. Who can explain why this figure shows two congruent triangles given the fact that AB is an angle bisector. A B C D

CONGRUENCE There are a few types of proofs. We’ll focus on two. Informal Proofs are a way of explaining why something is true in a paragraph form. Two-Column Proofs are a way of explaining why something is true in a more structured method.

CONGRUENCE Proofs are a step by step process that shows why something is true. The first step to writing a proof, is to write the given. This will usually be given to you…hence the name. A B C D

STATEMENT 1.AB is an angle bisector and AC is congruent to AD. JUSTIFICATION 1.Given

CONGRUENCE The next step is finding another statement(s) I can make, preferably one that gets us a step closer to one of the theorems. A B C D

STATEMENT 1.AB is an angle bisector and AC is congruent to AD. 2.AB is congruent to AB 3.Angle CAB is congruent to Angle DAB JUSTIFICATION 1.Given 2.Reflexive Property 3.Definition of Bisector

CONGRUENCE Finally, when we have enough information, we state what we wanted to prove and the theorem we can NOW use to prove it. A B C D

STATEMENT 1.AB is an angle bisector and AC is congruent to AD. 2.AB is congruent to AB 3.Angle CAB is congruent to Angle DAB 4.Triangle ABC is congruent to Triangle ABD JUSTIFICATION 1.Given 2.Reflexive Property 3.Definition of Bisector 4.SAS Congruence Postulate

CONGRUENCE An informal proof may read like this… We have been given the fact that AB is an angle bisector and AC is congruent to AD. We know that AB is congruent to AB due to the reflexive property. We also know that angle CAB is congruent to angle DAB due to the definition of angle bisector. Since all this is true, we know that triangle ABC is congruent to triangle ABD because of the SAS congruence postulate.

CONGRUENCE When giving justifications there are very few postulates and properties you will use. Get out your practice problems and we will go over a few now.

CONGRUENCE The big few will be these… – DEFINITION OF MIDPOINT – DEFINITION OF BISECTOR – DEFINITION OF VERTICAL ANGLES – REFLEXIVE PROPERTY I’ve made you a list of pretty much all the theorems you’ll ever need!