Overview
This lesson covers fundamental geometry concepts, including lines, angles, triangle properties, congruence postulates, and methods for proving triangles congruent.
Basic Geometric Elements
- A line extends infinitely in both directions and can be named by any two points on it (e.g., line AB).
- A ray starts at one point and extends infinitely in one direction (named from the endpoint, e.g., ray AB).
- A segment has two endpoints and is named by its endpoints (e.g., segment AB).
Angles and Their Types
- An acute angle measures greater than 0° but less than 90°.
- A right angle measures exactly 90°.
- An obtuse angle measures greater than 90° but less than 180°.
- A straight angle measures exactly 180° and forms a straight line.
Midpoint, Bisectors, and Related Terms
- A midpoint divides a segment into two congruent segments.
- A segment bisector (often a ray or line) passes through a segment’s midpoint, dividing it into two equal parts.
- An angle bisector is a ray that divides an angle into two equal angles.
Parallel and Perpendicular Lines
- Parallel lines never intersect and have the same slope; denoted by “∥”.
- Perpendicular lines intersect at 90°, with slopes that are negative reciprocals; denoted by “⟂”.
Angle Relationships
- Complementary angles sum to 90°.
- Supplementary angles sum to 180°.
- Vertical angles are formed by intersecting lines and are congruent.
- The transitive property: if a = b and c = b, then a = c.
Triangle Properties
- A median connects a triangle’s vertex to the midpoint of the opposite side.
- An altitude is a segment from a vertex perpendicular to the opposite side.
- A perpendicular bisector intersects a segment at its midpoint at a right angle.
Triangle Congruence
- SSS Postulate: All three sides of two triangles are congruent.
- SAS Postulate: Two sides and the included angle are congruent.
- ASA Postulate: Two angles and the included side are congruent.
- AAS Postulate: Two angles and a non-included side are congruent.
- CPCTC: Corresponding parts of congruent triangles are congruent.
Proving Triangle Congruence (Examples)
- Use given congruent sides/angles, shared sides (reflexive property), vertical angles, and congruence postulates to prove triangle congruence.
- Once triangles are congruent, corresponding sides and angles are congruent by CPCTC.
Key Terms & Definitions
- Line — extends infinitely in both directions.
- Ray — starts at one point and extends infinitely in one direction.
- Segment — part of a line with two endpoints.
- Vertex — common endpoint of angle rays.
- Congruent — exactly equal in size and shape.
- Bisector — divides something into two equal parts.
- Median — vertex to midpoint segment in a triangle.
- Altitude — vertex to perpendicular segment in a triangle.
- Perpendicular Bisector — line dividing a segment into two equal parts at 90°.
- CPCTC — corresponding parts of congruent triangles are congruent.
Action Items / Next Steps
- Review practice problems on triangle proofs and congruence.
- Study definitions and examples of geometric terms.
- Check supplemental videos or playlists for additional practice.