Triangle Congruencies HOMEWORK: Proofs WS & Triangle Packet - evens Quiz Friday January11, 2012 Lessons 4.3 – 4.5 Triangle Sum, Triangle Inequalities, Side-Angle, Triangle Congruency

Two Triangles are congruent if they are the same shape and size. To PROVE two triangles congruent means to show that all corresponding parts are congruent. Each triangle has 6 parts: 3 sides & 3 angles Congruent Triangles

However, shortcuts have been discovered and they make our job easier. SSS AAA SAS SSA ASA SAA Possible Triangle Congruency Conjecture Combinations S – SIDE A – ANGLE ORDER MATTERS!! The order of the corresponding parts in the triangles must mimic the order of the letters in the Congruency Statements

Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides SSS AAA SAS SSA ASA SAA Possible Triangle Congruency Combinations

Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides YES! SSS √ AAA SAS SSA ASA SAA

90° 65° 25° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles SSS AAA SAS SSA ASA SAA √

90° 65° 25° 90° 65° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles NO!

All Equilateral/Equiangular triangles have three 60° angles. 60° A Counterexample to an AAA congruence conjecture

SSS AAA SAS SSA ASA SAA √ Notes - Congruent Triangles Are these guarantees of triangle congruence? So far this is what has been determined from the list Now let’s try combinations with 2 sides and an angle

25° SSS AAA SAS SSA ASA SAA √ Notes - Congruent Triangles Is this a guarantee of triangle congruence? One pair of congruent angles, two pairs of congruent sides 25°

√ SSS AAA SAS SSA ASA SAA √ 25° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and the included angle YES! SAS! An included angle is between the two given sides.

A B C ABCDEF 92° D E F Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and the included angle √ SSS AAA SAS SSA ASA SAA √

65° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle √ SSS AAA SAS SSA ASA SAA √

65° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle √ SSS AAA SAS SSA ASA SAA √

65° √ SSS AAA SAS SSA ASA SAA √ Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle

65° √ SSS AAA SAS SSA ASA SAA √ Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle

A B C D E F ABCDEF √ SSS AAA SAS SSA ASA SAA √ NO! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle

90° 25° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side √ SSS AAA SAS SSA ASA SAA √

90° 25° 90° 25° √ SSS AAA SAS SSA ASA SAA √ YES! ASA! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side

65° 55° A B C 65° 55° D E F ABCDEF ASA! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side √ SSS AAA SAS SSA ASA SAA √ √

90° 35° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the non-included side √ SSS AAA SAS SSA ASA SAA √ √

35° 90° 35° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the non-included side √ SSS AAA SAS SSA ASA SAA √ √ √ NO YES!

√ SSS AAA SAS SSA ASA SAA √ √ √ Notes - Congruent Triangles These are the triangle congruencies. SSS SAS ASA SAA

Using Properties of Right triangles In Combination with Triangle Congruencies Hypotenuse –Leg Congruence Theorem (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. One More Congruency

HL (Hypotenuse - Leg) is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate. HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!

SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. Methods of Proving Triangles Congruent

Congruent Triangles Name the congruence ASA S F R 120° 35° D P Q 120° 35° FRSPQD Is?

SSA FRS QSR ? S F R 42° Q Congruent Triangles Name the congruence Shared Side – Reflexive Prop Is WHY NOT? SSA is NOT a valid Triangle congruence

SAS FRSQSR S F R 50° Q Congruent Triangles Name the congruence Shared Side – Reflexive Prop Is?

SSS ? MNRPTB M N R T P B Names of the triangles in the congruence statement are not in corresponding order. WHY NOT? Congruent Triangles Name the congruence Is?