Overview
This lesson covers foundational geometry concepts including types of lines, angles, triangle properties, congruence postulates, and important geometric relationships and proofs.
Lines, Rays, and Segments
- A line extends infinitely in both directions and is denoted with arrows.
- Lines can be named using any two points on the line.
- A ray starts at one point and extends infinitely in one direction.
- Rays are named starting with the endpoint and then another point on the ray.
- A segment has definite endpoints and is named by those endpoints.
Angles and Their Types
- An angle is formed by two rays sharing a common endpoint (vertex).
- Acute angle: measures greater than 0° and less than 90°.
- Right angle: exactly 90°.
- Obtuse angle: greater than 90° but less than 180°.
- Straight angle: exactly 180°.
Midpoints and Bisectors
- The midpoint divides a segment into two equal parts.
- A segment bisector passes through the midpoint, dividing a segment into two congruent segments.
- 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 a right angle (90°).
- Their slopes are negative reciprocals.
- Symbol: ⟂
Angle Relationships
- Complementary angles: two angles whose measures add up to 90°.
- Supplementary angles: two angles whose measures add up to 180°.
- Vertical angles: formed by two intersecting lines, opposite angles are congruent.
- Transitive property: if two things are each equal to a third thing, they are equal to each other.
Triangles: Medians, Altitudes, and Bisectors
- Median: segment from a vertex to the midpoint of the opposite side.
- Altitude: segment from a vertex perpendicular to the opposite side.
- Perpendicular bisector: a line perpendicular to a segment at its midpoint; every point on it is equidistant from the endpoints.
Triangle Congruence Postulates
- SSS (Side-Side-Side): all three sides congruent.
- SAS (Side-Angle-Side): two sides and the included angle congruent.
- ASA (Angle-Side-Angle): two angles and the included side congruent.
- AAS (Angle-Angle-Side): two angles and a non-included side congruent.
- CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
Proof Methods and Example Problems
- Use congruence postulates and properties (reflexive, vertical angles, etc.) to prove triangles and their parts are congruent.
- Proofs often involve identifying given congruencies, using bisectors or altitudes, and applying CPCTC.
Key Terms & Definitions
- Line — extends infinitely in two directions.
- Ray — starts at a point and extends infinitely in one direction.
- Segment — part of a line with two endpoints.
- Midpoint — divides a segment into two equal parts.
- Bisector — divides a segment or angle into two equal parts.
- Complementary angles — sum to 90°.
- Supplementary angles — sum to 180°.
- Vertical angles — opposite angles formed by intersecting lines, always congruent.
- Median — vertex to midpoint segment in a triangle.
- Altitude — vertex to perpendicular of the opposite side.
- Perpendicular bisector — perpendicular and passes through midpoint of segment.
- Congruent triangles — triangles with all corresponding parts equal.
- CPCTC — Corresponding parts of congruent triangles are congruent.
Action Items / Next Steps
- Review and memorize the four main triangle congruence postulates and key definitions.
- Practice identifying types of angles, segments, and lines in diagrams.
- Prepare for proofs by practicing with triangle congruence and angle relationships.