6.4 Pairs of Angles

Description

Quick Overview

This section covers the relationships and properties of different pairs of angles formed by intersecting lines and rays.

Standard

In Section 6.4, we explore pairs of angles, including complementary, supplementary, adjacent, and linear pairs. The section introduces two axioms and one theorem regarding these angles, elaborating on their definitions and relationships through examples and proofs.

Detailed

Pairs of Angles

In this section, we analyze various pairs of angles that arise when a ray stands on a line and when two lines intersect. We start by recalling the definitions of complementary and supplementary angles, alongside adjacent angles and linear pairs. We then examine what occurs when a ray stands on a line through geometric representations.

Key Points Covered:

  1. Angle Relations: We define how the angles formed by a ray on a line are related:
  2. If \( ext{ray OC}\) stands on line \( ext{AB}\), the angles formed (\( ext{∠ AOC, ∠ BOC, ∠ AOB}\)) satisfy \(∠ AOC + ∠ BOC = ∠ AOB\).
  3. If \(∠ AOB = 180°\), we conclude \(∠ AOC + ∠ BOC = 180°\) (Linear pair axiom).
  4. Axioms: Two critical axioms related to pairs of angles are introduced:
  5. Axiom 6.1: If a ray stands on a line, the sum of the two adjacent angles formed is 180°.
  6. Axiom 6.2: If the sum of two adjacent angles is 180°, then the non-common arms form a line.
  7. Theorem: We present a theorem regarding vertically opposite angles:
  8. Theorem 6.1: When two lines intersect, the vertically opposite angles are equal.
  9. Examples and Exercises: The concept is reinforced through various examples that illustrate how to apply these definitions, axioms, and theorems in practical problem-solving scenarios.

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Key Concepts

  • Linear Pair: A pair of adjacent angles that sum to 180°.

  • Vertically Opposite Angles: Angles opposite each other at an intersection, which are equal.

Memory Aids

🎵 Rhymes Time

  • When angles are complementary, they dance to ninety's tune; for supplementary, it's a straight line in afternoon.

📖 Fascinating Stories

  • Once upon a time, two angles met at a street corner. They realized when they worked together, they formed straight and right relations.

🎯 Super Acronyms

C, S for Complementary and Supplementary angles.

Examples

  • {'example': 'If two angles are complementary, and one angle measures 30°, what is the measurement of the other?', 'solution': 'Let the unknown angle be x. We know that x + 30° = 90°. Thus, x = 90° - 30° = 60°.'}

  • {'example': 'If line AB and line CD intersect, and angles ∠ AOB = 120°, what is ∠ BOC?', 'solution': 'Since ∠ AOB and ∠ BOC are vertically opposite angles, ∠ BOC = ∠ AOB = 120°.'}

Glossary of Terms

  • Term: Complementary Angles

    Definition:

    Two angles whose sum is 90°.

  • Term: Supplementary Angles

    Definition:

    Two angles whose sum is 180°.

  • Term: Adjacent Angles

    Definition:

    Two angles that have a common vertex and a common arm and do not overlap.

  • Term: Linear Pair of Angles

    Definition:

    A pair of adjacent angles whose non-common arms form a straight line.

  • Term: Vertically Opposite Angles

    Definition:

    The angles opposite each other when two lines intersect; they are equal.

  • Term: Ray

    Definition:

    A part of a line that starts at a point and extends infinitely in one direction.

  • Term: Angle

    Definition:

    Formed by two rays with a common endpoint called the vertex.