Essential Question: What do I need to know about two triangles before I can say they are congruent?

The Idea of Congruence Two geometric figures with exactly the same size and shape. A C B D E F

How much do you need to know. . . . . . about two triangles to prove that they are congruent?

Corresponding Parts Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C AB  DE BC  EF AC  DF  A   D  B   E  C   F ABC   DEF E D F

Do you need all six ? NO ! SSS SAS ASA AAS HL

The triangles are congruent by SSS. Side-Side-Side (SSS) If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. B A C Side E Side F D Side AB  DE BC  EF AC  DF ABC   DEF The triangles are congruent by SSS.

side-angle-side, or just SAS. Included Angle The angle between two sides GIH I GHI H HGI G This combo is called side-angle-side, or just SAS.

The other two angles are the NON-INCLUDED angles. Name the included angle: YE and ES ES and YS YS and YE S Y E  YES or E  YSE or S  EYS or Y The other two angles are the NON-INCLUDED angles.

Side-Angle-Side (SAS) The triangles are congruent by SAS. If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. included angle B E Side F A C Side D AB  DE A   D AC  DF Angle ABC   DEF The triangles are congruent by SAS.

angle-side-angle, or just ASA. Included Side The side between two angles GI GH HI This combo is called angle-side-angle, or just ASA.

The other two sides are the NON-INCLUDED sides. Name the included side: Y and E E and S S and Y S Y E YE ES SY The other two sides are the NON-INCLUDED sides.

Angle-Side-Angle (ASA) The triangles are congruent by ASA. If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. included side B E Angle Side F A C D Angle A   D AB  DE  B   E ABC   DEF The triangles are congruent by ASA.

Angle-Angle-Side (AAS) The triangles are congruent by AAS. If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. Non-included side B A C Angle E D F Side Angle A   D  B   E BC  EF ABC   DEF The triangles are congruent by AAS.

Warning: No SSA Postulate There is no such thing as an SSA postulate! Side Angle Side The triangles are NOTcongruent!

There is no such thing as an SSA postulate! Warning: No SSA Postulate There is no such thing as an SSA postulate! NOT CONGRUENT!

If we know that the two triangles are right triangles! BUT: SSA DOES work in one situation! If we know that the two triangles are right triangles! Side Side Side Angle

These triangles ARE CONGRUENT by HL! We call this HL, for “Hypotenuse – Leg” Remember! The triangles must be RIGHT! Hypotenuse Hypotenuse Leg RIGHT Triangles! These triangles ARE CONGRUENT by HL!

The triangles are congruent by HL. Hypotenuse-Leg (HL) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Right Triangle Leg Hypotenuse AB  HL CB  GL C and G are rt.  ‘s ABC   DEF The triangles are congruent by HL.

There is no such thing as an AAA postulate! Warning: No AAA Postulate There is no such thing as an AAA postulate! Different Sizes! Same Shapes! E B A C F D NOT CONGRUENT!

Congruence Postulates and Theorems SSS SAS ASA AAS AAA? SSA? HL

Name That Postulate (when possible) SAS ASA SSA AAS Not enough info!