Relationships between Angles

In this chapter, we will explore relationships between angles that are essential for solving geometry problems. You will learn about complementary angles (angles whose measures sum to $90^\circ$, forming a right angle), supplementary angles (angles whose measures sum to $180^\circ$, forming a straight angle), opposite angles at a vertex (also called vertically opposite angles, formed by intersecting lines), and angles created by parallel lines and a transversal, including corresponding, alternate, and co-interior angles. These concepts build on your understanding of right angles ($90^\circ$), straight angles ($180^\circ$), and full angles ($360^\circ$). These special angle relationships will help you recognise parallel lines and calculate unknown angles in geometric figures.

Complementary and Supplementary Angles

Definition Complementary

Two angles are complementary if the sum of their measures is $90^\circ$. Complementary angles together form a right angle, like the corner of a square.

Example

Calculate the measure of the unknown angle $x^\circ$ if it is complementary to a $35^\circ$ angle.

Answer

The sum of the measures of complementary angles is $90^\circ$.$$\begin{aligned}x^\circ + 35^\circ &= 90^\circ \\ x^\circ &= 90^\circ - 35^\circ \quad (\text{subtract } 35^\circ) \\ &= 55^\circ\end{aligned}$$

Definition Supplementary

Two angles are supplementary if the sum of their measures is $180^\circ$. Supplementary angles together form a straight angle (a straight line).

Example

Calculate the measure of the unknown angle $x^\circ$ if it is supplementary to a $110^\circ$ angle.

Answer

The sum of the measures of supplementary angles is $180^\circ$.$$\begin{aligned}x^\circ + 110^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 110^\circ \quad (\text{subtract } 110^\circ) \\ &= 70^\circ\end{aligned}$$

Opposite Angles at a Vertex

Definition Opposite Angles at a Vertex

Opposite angles at a vertex (also called vertically opposite angles) are pairs of angles that lie opposite each other when two lines intersect. They share a common vertex.

Proposition Equality of Opposite Angles at a Vertex

Opposite angles at a vertex are equal.

$\textcolor{colordef}{a^\circ} = \textcolor{colorprop}{b^\circ}$

Proof

Consider two lines intersecting at a point, forming opposite angles $a^\circ$ and $b^\circ$, and an adjacent angle $c^\circ$.

Since $a^\circ$ and $c^\circ$ form a straight angle:$$\textcolor{colordef}{a^\circ} + \textcolor{olive}{c^\circ} = 180^\circ \quad (\text{straight angle})$$Similarly, $b^\circ$ and $c^\circ$ form a straight angle:$$\textcolor{colorprop}{b^\circ} + \textcolor{olive}{c^\circ} = 180^\circ \quad (\text{straight angle})$$Thus, $\textcolor{colordef}{a^\circ} + \textcolor{olive}{c^\circ} = \textcolor{colorprop}{b^\circ} + \textcolor{olive}{c^\circ}$. Subtracting $\textcolor{olive}{c^\circ}$ from both sides gives:$$\textcolor{colordef}{a^\circ} = \textcolor{colorprop}{b^\circ}$$

Example

Calculate the measure of the unknown angle $x^\circ$.

Answer

Opposite angles at a vertex are equal.$$\begin{aligned}x^\circ &= 70^\circ \quad (\text{opposite angles are equal})\end{aligned}$$

Corresponding, Alternate, and Co-interior Angles

Definition Corresponding, Alternate, and Co-interior Angles

Corresponding angles are on the same side of a transversal that intersects two lines and are in matching corners relative to the intersected lines, like the top-right corners of each intersection.
Alternate angles (alternate interior angles) are on opposite sides of a transversal and lie between the two intersected lines, forming a "Z" shape.
Co-interior angles (also called consecutive interior angles) are on the same side of a transversal and between the two intersected lines, forming a "C" shape.

Example

Identify the following for the given diagram:

The corresponding angles.
The alternate angles.
The co-interior angles.

Answer

Corresponding angles: $\textcolor{colorprop}{a^\circ}$ and $\textcolor{purple}{e^\circ}$, $\textcolor{colordef}{b^\circ}$ and $\textcolor{green}{f^\circ}$, $\textcolor{olive}{c^\circ}$ and $\textcolor{teal}{g^\circ}$, $\textcolor{yellow}{d^\circ}$ and $\textcolor{orange}{h^\circ}$.
Alternate angles: $\textcolor{olive}{c^\circ}$ and $\textcolor{purple}{e^\circ}$, $\textcolor{yellow}{d^\circ}$ and $\textcolor{green}{f^\circ}$.
Co-interior angles: $\textcolor{olive}{c^\circ}$ and $\textcolor{green}{f^\circ}$, $\textcolor{yellow}{d^\circ}$ and $\textcolor{purple}{e^\circ}$.

Properties of Parallel Lines

Proposition Properties of Parallel Lines

If two lines are parallel and intersected by a transversal, then:

Corresponding angles are equal.
Alternate angles are equal.
Co-interior angles are supplementary (their measures sum to $180^\circ$).

Example

Calculate the measure of the unknown angle $x^\circ$, given that the lines are parallel.

Answer

Since the angles are alternate and the lines are parallel, they are equal.$$\begin{aligned}x^\circ &= 70^\circ \quad (\text{alternate angles are equal})\end{aligned}$$

Proposition Parallel Lines from Equal Angles

If two corresponding angles or two alternate angles formed by the same transversal are equal, then the lines are parallel.

Example

Show that the lines are parallel, given the angle measures.

Answer

Since the corresponding angles are equal ($110^\circ = 110^\circ$), the lines are parallel.