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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 Congruent Triangles 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 However, shortcuts have been discovered and they make our job easier. Possible Triangle Congruency Conjecture Combinations S – SIDE A – ANGLE SSS AAA SAS SSA ASA SAA 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 Possible Triangle Congruency Combinations SSS AAA SAS SSA ASA SAA Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides √SSS AAA SAS SSA ASA SAA YES! SSS Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles √SSS AAA SAS SSA ASA SAA 90° 25° 65° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles NO! 65° 65° 90° 90° 25° 25° All Equilateral/Equiangular triangles have three 60° angles. 60° 60° 60° 60° 60° 60° 60° 60° A Counterexample to an AAA congruence conjecture 60° Notes - Congruent Triangles Are these guarantees of triangle congruence? So far this is what has been determined from the list √SSS AAA SAS SSA ASA SAA Now let’s try combinations with 2 sides and an angle Notes - Congruent Triangles Is this a guarantee of triangle congruence? One pair of congruent angles, two pairs of congruent sides √SSS AAA SAS SSA ASA SAA 25° 25° 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 25° YES! SAS! An included angle is between the two given sides. 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 D A 92° 92° B C ABC DEF F E 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° 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° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle E √SSS AAA √SAS SSA ASA SAA ABC D DEF F NO! 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 25° 90° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side 25° √SSS AAA √SAS SSA ASA SAA 90° 25° 90° YES! ASA! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side E B √SSS AAA √SAS SSA √ASA SAA A 65° 55° 55° 65° D C ABC DEF ASA! F 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 90° 35° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the non-included side 35° √SSS AAA √SAS SSA √ASA √SAA YES! 35° NO 35° NO 90° Notes - Congruent Triangles These are the triangle congruencies. SSS SAS ASA SAA √SSS AAA √SAS SSA √ASA √SAA One More Congruency 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. 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!!!!! Methods of Proving Triangles Congruent 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. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. ASA AAS HL Congruent Triangles Name the congruence Is FRS PQD ? R P F 120° 120° 35° 35° D S ASA Q Congruent Triangles Name the congruence Is FRS QSR ? R 42° F Q 42° Shared Side – Reflexive Prop S SSA WHY NOT? SSA is NOT a valid Triangle congruence Congruent Triangles Name the congruence Is FRS QSR ? R 50° F Q 50° Shared Side – Reflexive Prop S SAS Congruent Triangles Name the congruence MNR Is R PTB ? B N P M SSS ? T WHY NOT? Names of the triangles in the congruence statement are not in corresponding order.
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