Basic Concepts in Geometry

Lines

Line: A straight path extending in both directions forever.
- Represented with two arrows.
- Example: Line AB, Line BC, Line AC

Rays

Ray: A line with a starting point and extends in one direction forever.
- Example: Ray AC, Ray AB (starting point must be the initial point)

Segments

Segment: A part of a line with a beginning and end.
- Example: Segment AB

Angles

Acute Angle: Measure between 0 to 90 degrees.
- Example: Angle ABC (formed by rays BA and BC)
Right Angle: Measure of 90 degrees.
Obtuse Angle: Measure greater than 90 degrees but less than 180.
Straight Angle: Measure of 180 degrees, looks like a straight line.

Midpoint

Midpoint: Point exactly in the middle of a segment.
- Example: If B is the midpoint of segment AC, then AB is congruent to BC.

Segment Bisector

Segment Bisector: A line or ray that cuts another segment into two equal parts at its midpoint.
- Example: Ray RB bisects segment AC at point B.

Angle Bisector

Angle Bisector: A ray that divides an angle into two equal parts.
- Example: Ray BD bisects angle ABC into two equal parts. If angle ABC = 60 degrees, then angle ABD and angle DBC are 30 degrees each.

Parallel and Perpendicular Lines

Parallel Lines: Lines that never intersect and have the same slope.
- Notation: A ∥ B
Perpendicular Lines: Lines that intersect at right angles (90 degrees).
- Notation: A ⊥ B
- Slopes are negative reciprocals of each other.

Complementary and Supplementary Angles

Complementary Angles: Two angles that add up to 90 degrees.
- Example: Angle A (40 degrees) and Angle C (50 degrees).
Supplementary Angles: Two angles that add up to 180 degrees.
- Example: Angle ABD (110 degrees) and Angle DBC (70 degrees).

Transitive Property

Transitive Property: If two angles are congruent to the same angle, they are congruent to each other.
- Example: If Angle 1 ≅ Angle 2 and Angle 3 ≅ Angle 2, then Angle 1 ≅ Angle 3.

Vertical Angles

Vertical Angles: Angles formed by two intersecting lines, opposite angles are congruent.
- Example: Angle 1 ≅ Angle 3, Angle 2 ≅ Angle 4.

Medians and Altitudes

Median: A line segment from a vertex of a triangle to the midpoint of the opposite side, splitting it into two equal parts.
- Example: Segment BD from vertex B to midpoint D of segment AC.
Altitude: A line segment from a vertex of a triangle perpendicular to the opposite side, forming right angles.
- Example: Segment BD forms right angles with segment AC.
- Notation: Segment BD ⊥ Segment AC

Perpendicular Bisectors

Perpendicular Bisector: A line that bisects a segment into two equal parts and is perpendicular to it.
- Notation: Angle AMQ ≅ Angle BMQ, Segment AM ≅ Segment BM
- Example: Line L is perpendicular to segment AB and bisects it at M.

Proving Triangle Congruence

Four key postulates:
- SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of another triangle.
- SAS (Side-Angle-Side): If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle.
- 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.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle.
- CPCTC: Once two triangles are proven congruent, Corresponding Parts of Congruent Triangles are Congruent.

Additional Example Problems:

Given Segment AD ≅ Segment DC and Segment AB ≅ Segment BC, prove Triangle ABD ≅ Triangle CBD using SSS postulate.
Prove Angle B ≅ Angle D if Angle A ≅ Angle E and Segment AC ≅ Segment CE using vertical angles and ASA postulate.
Given BD is an altitude and Angle A ≅ Angle C, prove Triangle ABD ≅ Triangle CBD using reflexive property and CPCTC.

For additional problems and detailed proofs, check out the links provided in the descriptions of relevant videos and don't forget to subscribe for updates!