Overview
This lesson covers essential geometry concepts, including lines, angles, congruence postulates, bisectors, medians, altitudes, and properties of triangles and angles.
Lines, Rays, and Segments
- A line extends infinitely in both directions and is named by any two points on it.
- A ray starts at a point and extends infinitely in one direction; the starting point must be named first.
- A segment has two endpoints and does not extend infinitely.
Types of Angles
- Acute angle: less than 90°.
- Right angle: exactly 90°.
- Obtuse angle: greater than 90° but less than 180°.
- Straight angle: exactly 180°, appears as a line.
Midpoints and Bisectors
- The midpoint of a segment divides it into two equal parts.
- A segment bisector passes through the midpoint, splitting 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 equal slopes; symbol: ‖.
- Perpendicular lines intersect at right angles; their slopes are negative reciprocals; symbol: ⟂.
Complementary and Supplementary Angles
- Complementary angles sum to 90°.
- Supplementary angles sum to 180°.
Properties and Theorems
- Transitive property: If angle 1 ≅ angle 2 and angle 3 ≅ angle 2, then angle 1 ≅ angle 3.
- Vertical angles are formed by intersecting lines; opposite angles are congruent.
Triangle-Related Concepts
- A median connects a vertex to the midpoint of the opposing side, dividing the side equally.
- An altitude connects a vertex perpendicularly to the opposite side, forming right angles.
- A perpendicular bisector bisects a segment at a right angle and any point on it is equidistant from the segment’s endpoints.
Triangle Congruence Postulates
- SSS (Side-Side-Side): All corresponding sides 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.
Key Terms & Definitions
- Congruent (≅): Equal in shape and size.
- Bisector: A line or ray that 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.
- Perpendicular Bisector: A line that both bisects a segment and is perpendicular to it.
- CPCTC: After proving triangles congruent, corresponding parts are also congruent.
Action Items / Next Steps
- Review the four triangle congruence postulates and practice identifying them in triangle problems.
- Practice problems related to bisectors, medians, altitudes, and properties of angles.
- Check assigned readings or provided video playlists for more practice and detailed proofs.