Triangle Congruencies Lesson 4.4

c)What is PZ? d)What is <Z e)What is < N f)What percent of M&Ms are brown? 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? Given PQZ CNB Z P Q C N B ° 62°

Isosceles Triangle Conjecture If a triangle is isosceles, then base angles are equal. 80° 20° Isosceles Triangles Conjecture

Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles. 20° 25°45° Exterior Angle Conjecture

Classwork Review GEA NCA

IEC JAN Classwork Review

Lesson 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

90° 65° 25° Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles

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

SSS AAA SAS SSA ASA SAA √ Notes - Congruent Triangles Are these guarantees of triangle congruence?

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!

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 √ An included angle is between the two given sides.

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 √ SSS AAA SAS SSA ASA SAA √

A B C D E F ABCDEF 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 √ NO!

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° 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 √ YES! ASA!

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 √ √ √

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

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 FRSPQD S F R 120° 35° D P Q 120° 35°

SSA FRS QSR S F R 42° Q Congruent Triangles Name the congruence

SAS FRSQSR S F R 50° Q Congruent Triangles Name the congruence

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

SSS ASA