Line: Extends infinitely in both directions, represented by arrows on both ends.
Can be named based on the points it passes through (e.g., Line AB, Line BC, Line AC).
Rays
Ray: Extends infinitely in one direction from a single point.
Named starting with the first point (e.g., Ray AB, Ray AC, but not Ray BC).
Segments
Segment: Has a definite start and end point.
Named by its endpoints (e.g., Segment AB, Segment BA).
Angles
Acute Angle: Measures between 0 and 90 degrees.
Right Angle: Measures exactly 90 degrees.
Obtuse Angle: Measures between 90 and 180 degrees.
Straight Angle: Measures exactly 180 degrees.
Midpoint
Midpoint: Divides a segment into two equal parts.
If B is the midpoint of Segment AC, then Segment AB ≅ Segment BC.
Bisectors
Segment Bisector: A line, ray or segment that divides a line segment into two equal parts.
Example: If Ray RB passes through the midpoint of Segment AC, then AB ≅ BC.
Angle Bisector: A ray that divides an angle into two equal angles.
Example: If Ray BD bisects ∠ABC, then ∠ABD = ∠DBC.
Parallel and Perpendicular Lines
Parallel Lines: Never intersect and have equal slopes.
Example: If Line A and Line B have the same slope, they are parallel (A ∥ B).
Perpendicular Lines: Intersect at right angles.
Example: If Line A is perpendicular to Line B, and the slope of B is 2/5, the slope of A is -5/2 (A ⊥ B).
Complementary and Supplementary Angles
Complementary Angles: Two angles that add up to 90 degrees.
Example: ∠A = 40° and ∠C = 50° are complementary (∠A + ∠C = 90°).
Supplementary Angles: Two angles that add up to 180 degrees.
Example: ∠ABD = 110° and ∠DBC = 70° are supplementary (∠ABD + ∠DBC = 180°).
Transitive Property
If two angles are congruent to the same angle, they are congruent to each other.
Example: If ∠1 ≅ ∠2 and ∠3 ≅ ∠2, then ∠1 ≅ ∠3.
Vertical Angles
Vertically opposite angles formed by two intersecting lines are congruent.
Example: If ∠1 = 50°, then ∠3 = 50°.
Linear pairs add up to 180°.
Medians and Altitudes
Median: A segment from the vertex to the midpoint of the opposite side in a triangle.
Example: In ∆ABC, if D is the midpoint of AC, then BD is a median.
Altitude: A segment from the vertex perpendicular to the opposite side.
Example: In ∆ABC, if BD forms a right angle with AC, then BD is an altitude.
Perpendicular Bisectors
Combines properties of medians and altitudes: bisects a segment into two equal parts and forms right angles.
Example: In Segment AB, if Line L is a perpendicular bisector, then AM ≅ BM and ∠AMP = ∠BMP = 90°.
Proving Triangle Congruence
SSS Postulate: If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
Triangle ABC ≅ Triangle DEF.
SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
Triangle XYZ ≅ Triangle RST.
ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
Triangle MNO ≅ Triangle GHI.
AAS Postulate: If two angles and a non-included side are congruent between two triangles, then the triangles are congruent.
Triangle ABC ≅ Triangle RST.
CPCTC (Corresponding Parts of Congruent Triangles are Congruent): Once two triangles are proven congruent, their corresponding parts are congruent.
Example Proofs
Prove triangle congruence using given criteria and postulates (SSS, SAS, ASA, AAS).
Use reflexive property where triangles share a common side.
Use vertical angles and CPCTC to determine congruent angles or sides.