Overview
This lecture introduces foundational geometry concepts, including types of lines, angles, triangle properties, congruence postulates, and properties necessary for geometric proofs.
Lines, Rays, and Segments
- A line extends infinitely in both directions and is named by any two points on it.
- A ray starts at one point and extends infinitely in one direction, named by its endpoint first.
- A segment has two endpoints and is named by its endpoints.
Angles and Their Types
- An angle is formed by two rays meeting at a common endpoint (vertex).
- An acute angle measures less than 90 degrees.
- A right angle measures exactly 90 degrees.
- An obtuse angle measures greater than 90 and less than 180 degrees.
- A straight angle measures exactly 180 degrees.
Midpoints, Bisectors, and Related Terms
- The midpoint of a segment divides it into two equal parts.
- A segment bisector passes through the midpoint and divides the segment into two congruent parts.
- An angle bisector divides an angle into two equal angles.
Parallel and Perpendicular Lines
- Parallel lines never intersect and have the same slope (symbol: ||).
- Perpendicular lines intersect at 90-degree angles; their slopes are negative reciprocals (symbol: ⟂).
Complementary and Supplementary Angles
- Complementary angles add up to 90 degrees.
- Supplementary angles add up to 180 degrees.
Properties and Relationships
- The transitive property: If two quantities are equal to a third, they are equal to each other.
- Vertical angles are the opposite angles formed by intersecting lines and are congruent.
Triangle Segments: Medians, Altitudes, Perpendicular Bisectors
- A median connects a vertex to the midpoint of the opposite side.
- An altitude is a segment from a vertex perpendicular to the opposite side.
- A perpendicular bisector is a line that is perpendicular to a segment at its midpoint and divides it into two equal parts; any point on it is equidistant from the segment's endpoints.
Triangle Congruence Postulates and Proofs
- SSS (Side-Side-Side): All three sides of two triangles are congruent.
- SAS (Side-Angle-Side): Two sides and the included angle are congruent.
- ASA (Angle-Side-Angle): Two angles and the included side are congruent.
- AAS (Angle-Angle-Side): Two angles and a non-included side are congruent.
- CPCTC: Corresponding Parts of Congruent Triangles are Congruent; used after proving triangle congruence.
- Reflexive property: A segment or angle is congruent to itself.
Key Terms & Definitions
- Line — infinite set of points extending in two directions.
- Ray — part of a line with one endpoint, extending infinitely in one direction.
- Segment — part of a line with two endpoints.
- Bisector — divides a segment or angle into two equal parts.
- Median — segment from a triangle’s vertex to the midpoint of the opposite side.
- Altitude — segment from a vertex perpendicular to the opposite side.
- Congruent — exactly equal in size and shape.
- Complementary angles — two angles whose measures add to 90°.
- Supplementary angles — two angles whose measures add to 180°.
- Transitive property — if a = b and b = c, then a = c.
- Vertical angles — opposite angles formed by two intersecting lines.
- CPCTC — corresponding parts of congruent triangles are congruent.
Action Items / Next Steps
- Review triangle congruence postulates and their application in proofs.
- Practice identifying and working with medians, altitudes, and perpendicular bisectors.
- Complete additional practice problems on geometry proofs as recommended in the video description.