Congruence
Two figures are congruent () if they have the exact same size and shape. If you cut one out, it would fit perfectly over the other.
Congruence Criteria
To prove , we typically need 3 pieces of information.
Theorem (SSS (Side-Side-Side))
If three sides of one triangle are equal to three sides of another triangle, they are congruent.
Theorem (SAS (Side-Angle-Side))
If two sides and the included angle of one triangle are equal to the corresponding parts of another, they are congruent.
Theorem (ASA (Angle-Side-Angle))
If two angles and the included side are equal, the triangles are congruent.
Theorem (AAS (Angle-Angle-Side))
If two angles and a non-included side are equal, the triangles are congruent.
Theorem (HL (Hypotenuse-Leg))
If the hypotenuse and one leg of a right triangle are equal to the corresponding parts of another right triangle, they are congruent.
Common Pitfall: SSA
SSA (Side-Side-Angle) is NOT a valid congruence criterion.- Exception: If the angle is (HL) or obtuse, SSA works.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent. Once you prove congruence, you get all other equalities for free.Practice Problems
Exercise (Problem 1)
Let be a parallelogram. Prove that diagonal divides it into two congruent triangles.
Exercise (Problem 2)
Given segments and bisecting each other at , prove that .
Exercise (Problem 3)
In acute , altitudes and have equal lengths. Prove that is isosceles.