Do-Now 2) Find the value of x & the measure of each angle. 5x – 4 4x ° 1) Find the value of x. 4x x – 10 3x + 1 5x – 4 + 4x + 14 = 100 9x + 10 = 100 9x = 90 x = 10 2x – x + 1 = 4x x – 9 = 4x + 12 x = 21 96° 32° 64°

Congruence Two geometric figures with exactly the same size and shape. A C B DE F How much do you need to know... How much do you need to know... about two triangles to prove that they are congruent?

 ABC   DEF B A C E D F 1.AB  DE 2.BC  EF 3.AC  DF 4.  A   D 5.  B   E 6.  C   F If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Corresponding Parts

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

Congruent Triangles Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If two sides and the included angle of one triangle are congruent to two sides and the included angle of second triangle, then the two triangles are congruent. Side-Angle-Side (SAS) Congruence Postulate

The angle between two sides Included Angle  G G  I I  H H

Name the included angle: YE and ES ES and YS YS and YE Included Angle SY E  E E  S S  Y Y

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Side-Angle (ASA) Congruence Postulate If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Theorem Examples

The side between two angles Included Side GI HI GH

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Side-Angle (ASA) Congruence Postulate If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Theorem Examples

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. Hypotenuse-Leg (HL) Congruence Theorem CPCTC Theorem Corresponding Parts of Congruent Triangles are Congruent This is used when proving that either a pair of angles or two sides are congruent. 1 st Prove that the triangles are congruent by SSS, SAS, ASA or AAS 2 nd Knowing that all corresponding parts of congruent triangles are congruent, you can then state that those angles or sides are congruent.

Warning: No SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!

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

Name That Postulate SAS ASA SSS SSA (when possible)

Name That Postulate (when possible) ASA SAS AAA SSA

Name That Postulate (when possible) SAS SAS SAS Reflexive Property Vertical Angles Reflexive Property SSA