Overview
This lecture covers essential geometry concepts, including types of lines, angles, triangle properties, bisectors, and proofs for triangle congruence.
Lines, Rays, and Segments
- A line extends infinitely in both directions and is denoted with arrows on both ends.
- A ray starts at a point and extends infinitely in one direction.
- A segment has two endpoints and does not extend infinitely.
Angles and Their Types
- An angle is formed by the union of two rays sharing a common endpoint (vertex).
- Acute angle: measures between 0° and 90°, exclusive.
- Right angle: measures exactly 90°.
- Obtuse angle: measures greater than 90°, less than 180°.
- Straight angle: measures exactly 180°, forms a straight line.
Midpoints, Bisectors, and Vertical Angles
- The midpoint divides a segment into two congruent (equal) parts.
- A segment bisector passes through the midpoint, dividing the segment into two equal parts.
- An angle bisector splits an angle into two equal angles.
- Vertical angles are formed by intersecting lines and are always congruent to each other.
Parallel and Perpendicular Lines
- Parallel lines never intersect and have equal slopes.
- Perpendicular lines intersect at right angles; their slopes are negative reciprocals.
Complementary and Supplementary Angles
- Complementary angles: two angles whose measures sum to 90°.
- Supplementary angles: two angles whose measures sum to 180°.
Properties and Postulates in Triangles
- Median: a segment from a triangle’s vertex to the midpoint of the opposite side.
- Altitude: a segment from a vertex perpendicular to the opposite side.
- Perpendicular bisector: a line that bisects a segment at 90° and passes through its midpoint.
- Any point on a perpendicular bisector is equidistant from the segment’s endpoints.
Triangle Congruence and Proofs
- SSS Postulate: triangles are congruent if all three sides are congruent.
- SAS Postulate: triangles are congruent if two sides and the included angle are congruent.
- ASA Postulate: triangles are congruent if two angles and the included side are congruent.
- AAS Postulate: triangles are congruent if two angles and a non-included side are congruent.
- CPCTC: corresponding parts of congruent triangles are congruent.
- Reflexive property: a segment or angle is congruent to itself.
- Transitive property: if two angles (or segments) are congruent to the same angle (or segment), they are congruent to each other.
Key Terms & Definitions
- Line — infinite set of points extending in both directions.
- Ray — has a starting point and extends infinitely in one direction.
- Segment — part of a line between two endpoints.
- Midpoint — divides a segment into two equal parts.
- Bisector — divides an angle or segment into two equal parts.
- Median — segment from vertex to midpoint of the opposite side in a triangle.
- Altitude — segment from vertex perpendicular to the opposite side.
- Vertical Angles — opposite angles formed by intersecting lines; always equal.
- Complementary Angles — two angles summing to 90°.
- Supplementary Angles — two angles summing to 180°.
- Congruent — exactly equal in measure or length.
Action Items / Next Steps
- Review geometry video playlists for additional practice problems.
- Study two-column proofs for triangle congruence.
- Practice identifying and using the SSS, SAS, ASA, and AAS postulates.