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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.
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.
Linear Pair: A pair of adjacent angles that sum to 180°.
Vertically Opposite Angles: Angles opposite each other at an intersection, which are equal.
When angles are complementary, they dance to ninety's tune; for supplementary, it's a straight line in afternoon.
Once upon a time, two angles met at a street corner. They realized when they worked together, they formed straight and right relations.
{'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°.'}
Term: Complementary Angles
Definition: Two angles whose sum is 90°.
Two angles whose sum is 90°.
Term: Supplementary Angles
Definition: Two angles whose sum is 180°.
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.
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.
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.
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.
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.
Formed by two rays with a common endpoint called the vertex.