Geometry Basics Lecture Notes

Introduction

• Discussion on basic geometry concepts.
• Key topics: Lines, Rays, Segments, Angles, Midpoints, Bisectors, Parallel & Perpendicular Lines, Complementary & Supplementary Angles, Transitive Property, Vertical Angles, Medians, Altitudes, Perpendicular Bisectors, Triangle Congruence.

Lines

• Line: Extends in both directions forever with arrows on both ends.
  • Example: Line AB, can also be called Line BC or Line AC.

Rays

• Ray: Has a starting point and extends infinitely in one direction.
  • Naming: Start with the initial point; e.g., Ray AB (not BC if A is the starting point).

Segments

• Segment: Has a definite beginning and end.
  • Example: Segment AB or Segment BA.

Angles

• Acute Angle: Measures less than 90 degrees.
• Right Angle: Measures exactly 90 degrees.
• Obtuse Angle: Measures more than 90 but less than 180 degrees.
• Straight Angle: Measures 180 degrees.

Midpoint

• Midpoint divides a segment into two equal parts.
  • Segment AB is congruent to segment BC if B is the midpoint of AC.

Bisectors

• Segment Bisector: A line or ray that divides a segment into two equal parts.
• Angle Bisector: A ray that divides an angle into two equal parts.

Parallel Lines

• Parallel Lines: Never intersect and have the same slope.
  • Symbol: Line A ∥ Line B.

Perpendicular Lines

• Perpendicular Lines: Intersect at right angles (90 degrees).
  • Finding Slope: Slopes are negative reciprocals.

Complementary Angles

• Sum to 90 degrees.
  • Example: Angle A is 40 degrees, Angle C is 50 degrees (A and C are complementary).

Supplementary Angles

• Sum to 180 degrees.
  • Example: Angle ABD is 110 degrees, then Angle DBC is 70 degrees.

Transitive Property

• If two angles (or sides) are congruent to the same angle (or side), they are congruent to each other.
  • Example: If Angle 1 ≅ Angle 2 and Angle 3 ≅ Angle 2, then Angle 1 ≅ Angle 3.

Vertical Angles

• Formed by intersecting lines, opposite angles are congruent.

Medians

• A median connects a vertex of a triangle to the midpoint of the opposite side.

Altitudes

• An altitude is a perpendicular segment from a vertex to the line containing the opposite side.
  • Forms right angles within the triangle.

Perpendicular Bisector

• Combines properties of a median and an altitude.
  • Cuts a segment into two equal parts and forms right angles.

Triangle Congruence Postulates

1. SSS (Side-Side-Side) Postulate: If three sides of one triangle are congruent to three sides of another, the triangles are congruent.
2. SAS (Side-Angle-Side) Postulate: If two sides and the included angle of one triangle are congruent to another, the triangles are congruent.
3. ASA (Angle-Side-Angle) Postulate: If two angles and the included side of one triangle are congruent to another, the triangles are congruent.
4. AAS (Angle-Angle-Side) Postulate: If two angles and a non-included side of one triangle are congruent to another, the triangles are congruent.

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

• Once triangles are proven congruent, all their corresponding parts are congruent.

Practice and Further Learning

• Practice problems and additional resources available via linked playlists and supplementary videos.
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