Triangle Congruence Theorems Geometry Triangle Congruence Theorems

Congruent Triangles Congruent triangles have three congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.

The Triangle Congruence Postulates &Theorems AAS ASA SAS SSS FOR ALL TRIANGLES LA HA LL HL FOR RIGHT TRIANGLES ONLY

Theorem If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. Think about it… they have to add up to 180°.

A closer look... If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. 85° 30° But do the two triangles have to be congruent?

Example Why aren’t these triangles congruent? 30° 30° Why aren’t these triangles congruent? What do we call these triangles?

So, how do we prove that two triangles really are congruent?

the 2 triangles are CONGRUENT! ASA (Angle, Side, Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! AAS (Angle, Angle, Side) Special case of ASA If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! SAS (Side, Angle, Side) If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . F E D A C B then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! SSS (Side, Side, Side) F E D A C B In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! HL (Hypotenuse, Leg) If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B F E D then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! HA (Hypotenuse, Angle) F E D A C B If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! LA (Leg, Angle) A C B F E D If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

the 2 triangles are CONGRUENT! LL (Leg, Leg) A C B F E D If both pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides

Summary --- for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg HA – Pair of hypotenuses and one acute angle LL – Both pair of legs LA – One pair of legs and one pair of acute angles

Congruent triangles have 3 congruent sides and 3 congruent angles. Informal Geometry 6/2/2018 Congruent Triangles Congruent triangles have 3 congruent sides and 3 congruent angles. The parts of congruent triangles that “match” are called corresponding parts.

ORDER MATTERS!!!! In a congruence statement Everything matches up. Informal Geometry 6/2/2018 Congruence Statement In a congruence statement ORDER MATTERS!!!! Everything matches up.

Corresponding Parts of Congruent Triangles are Congruent CPCTC Corresponding Parts of Congruent Triangles are Congruent

Complete each congruence statement. B If ABC  DEF, then BC  ___ EF A C D F E

Complete each congruence statement. B If ABC  DEF, then A  ___ D A C D F E

Complete each congruence statement. B If ABC  DEF, then C  ___ F A C D F E

Fill in the blanks If CAT  DOG, then AC  ___ OD

BAT  MON N T  ___ _____  ONM _____  MO ATB NM  ____ BA TB Fill in the blanks BAT  MON N T  ___ _____  ONM _____  MO NM  ____ ATB BA TB

Fill in the blanks BCA   ____ ____   GFE EGF CAB

Complete the congruence statement. MKL _____   JKN

Complete the congruence statement. ABD _____   CBD

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