Angles: Adjacent, Supplementary and Vertically Opposite

Angles are formed when two lines meet at a point. Understanding different types of angle relationships helps us solve geometric problems. In this lesson, we'll focus on three important angle relationships: adjacent angles, supplementary angles, and vertically opposite angles.

Adjacent Angles

Adjacent angles share a common vertex and a common side, but do not overlap:

∠AOB and ∠BOC are adjacent angles

Vertically Opposite Angles

When two lines intersect, the angles opposite each other are called vertically opposite angles:

\[ \angle A = \angle B \text{ and } \angle C = \angle D \]

Finding Unknown Angles

We can use these angle relationships to find unknown angles in geometric figures:

Example 1: Adjacent Angles

If two adjacent angles form a right angle (90°), and one angle is 35°, find the other angle.

\[ \text{Let the angles be } \angle A \text{ and } \angle B \] \[ \angle A + \angle B = 90^\circ \] \[ 35^\circ + \angle B = 90^\circ \] \[ \angle B = 90^\circ - 35^\circ = 55^\circ \]

Example 2: Supplementary Angles

Two angles are supplementary. One angle is 112°. Find the other angle.

\[ \angle A + \angle B = 180^\circ \] \[ 112^\circ + \angle B = 180^\circ \] \[ \angle B = 180^\circ - 112^\circ = 68^\circ \]

Example 3: Vertically Opposite Angles

In the figure below, find the angles marked with letters:

Given: One angle is 75°

\[ a = 75^\circ \text{ (vertically opposite)} \] \[ b = 180^\circ - 75^\circ = 105^\circ \text{ (supplementary)} \] \[ c = 105^\circ \text{ (vertically opposite to } b) \]

Practical Examples

Understanding these angle relationships is useful in many real-world situations:

Construction and Architecture

Builders use angle relationships to ensure walls meet at correct angles:

Road Intersections

Vertically opposite angles help design proper road crossings:

Art and Design

Artists use angle relationships to create perspective and balance: