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Triangle Congruencies Lesson 4.4 Opener a) Can you make a triangle using the sides: 5 ft., 12 ft., and 4 ft.? b) One side of a triangle is 12 mm. Another side is 17 mm. What are the possible lengths for the third side? Q Given PQZ CNB 33° 3 c) What is PZ? d) What is <Z e) What is < N f) What percent of M&Ms are brown? Z 10 P B 62° N C Isosceles Triangles Conjecture Isosceles Triangle Conjecture If a triangle is isosceles, then base angles are equal. 20° 80° 80° Exterior Angle Conjecture Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles. 20° 45° 25° Classwork Review GEA NCA Classwork Review JAN IEC Lesson 4.4 - Congruent Triangles Is this a guarantee of triangle congruence? One pair of congruent sides Notes - Congruent Triangles Is this a guarantee of triangle congruence? One pair of congruent sides NO! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides NO! Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides YES! SSS Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles 90° 25° 65° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles NO! 65° 65° 90° 25° 90° 25° Notes - Congruent Triangles Are these guarantees of triangle congruence? √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 √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! 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° E 92° B C ABC DEF F 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 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 √SSS AAA √SAS SSA ASA SAA 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 35° 35° 90° Notes - Congruent Triangles These are the triangle congruencies. SSS SAS ASA SAA √SSS AAA √SAS SSA √ASA √SAA Using Properties of Right triangles Theorem 4.8 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. Using Properties of Right triangles 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 FRS PQD R P 120° F 120° 35° D 35° S ASA Q Congruent Triangles Name the congruence FRS QSR R 42° F 42° S SSA Q Congruent Triangles Name the congruence FRS QSR R 50° F 50° S SAS Q Congruent Triangles Name the congruence MNR PTB B R N SSS P M WHY? T Names of the triangles in the congruence statement are not in corresponding order. ASA SSS
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