Congruence

Third Angle Theorem: If two angles of one triangle are congruent to two angles of the other, their third angles are congruent.

Postulates/axioms to prove 2 triangles’ congruence: AAS (A= Angle, S= Side) ASA SAS SSS HL (H=Hypotenuse, L=Leg)

The following postulates/axioms do not work: ASS AAA

The order you read the angles and lines of the figures must match the order of the angles and lines in the postulates Ex: SAS - The figure must have an angle in between the 2 sides.

Geometric Proofs: Terms: Axiom - Something that is standardly accepted as true. It is accepted as fact and it makes sense/is self-evident Postulate - You assume this as true in order to prove something else. Theorem - A general proposition not self-evident but proved by a chain of reasoning Logical Argument- If it says something is true, your reason is “Given”

Two-column proofs: The first column is Statement, where you show your work. The second column is Reasons, where you tell why you can do that Example: Given x+y=60 and Given x=5, prove that y=55

9 most common Properties, Definitions, and Theorems for Triangles: Reflexive property: AB = BA When the triangles have an angle or side in common

Definition of midpoint Results in two segments being congruent

Vertical angles are congruent

Definition of angle bisector A line that divides an angle into 2 congruent angles

Right angles are congruent All right angles are congruent

Definition of perpendicular bisector Results in a 90-degree angle

Alternate interior angles of Parallel lines are congruent

3rd angles theorem If 2 corresponding angles are congruent the 3rd angle is also congruent.

Definition of segment bisector This is when 2 segments are equal due to a line bisecting the full line

Supplementary- 2 OR more angles make 180 degrees

Linear Pair - Form a line, Adjacent to each other, Share line and vertex. Supplementary

Complementary- 2 OR more angles make 90 degrees altogether

Vertical Angles - 2 angles intersect each other and the angles opposite from each other have the same measure

Transversal- A-line (never-ending on both sides), ray (one endpoint, one never-ending side) or segment (2 endpoints) that intersects 2 or more coplanar (same plane) lines, rays, or segments, each at different points. This creates alternate interior and exterior lines.

Interior and exterior angles - Used in transversal parallel lines. Interior is inside, exterior is outside.

Alternate interior theorem- If 2 lines cut by a transversal are parallel, then alternate interior angles are congruent.

 Alternate Exterior theorem- If two lines cut by a transversal are parallel, then alternate exterior angles are congruent.

Same side interior angles- If 2 lines cut by a transversal are parallel, then same side interior angles are supplementary.

Corresponding Angles Postulate- If 2 lines cut by a transversal are parallel, then corresponding angles are congruent.