Basic Terms and the Linear Pair

Before we prove anything, let's nail down the language. A geometric figure has a small bestiary of named objects , line, ray, segment, angle , and a few standard kinds of angles. After that, the headline result of the lesson is the linear pair: when two adjacent angles share an arm and the other arms make a straight line, the angles sum to $180^\circ$.

Definitions

• A line has no thickness and extends indefinitely in both directions. We write the line through $A$ and $B$ as $\overleftrightarrow{AB}$.
• A line segment $\overline{AB}$ is the portion of the line between $A$ and $B$, including $A$ and $B$.
• A ray $\overrightarrow{AB}$ starts at $A$ and goes through $B$, extending forever beyond $B$ in that direction.
• An angle is the figure formed by two rays sharing an endpoint, the vertex. We write $\angle ABC$ for the angle at vertex $B$ formed by rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$.
• An angle is acute if it measures between $0^\circ$ and $90^\circ$, right if exactly $90^\circ$, obtuse if between $90^\circ$ and $180^\circ$, straight if exactly $180^\circ$, and reflex if between $180^\circ$ and $360^\circ$.
• Two angles are complementary if they sum to $90^\circ$, and supplementary if they sum to $180^\circ$.
• Two angles are adjacent if they share a common arm (and a common vertex) and their interiors don't overlap.

The linear pair

A linear pair is a pair of adjacent angles whose non-common arms form a straight line. The defining property is:

Linear Pair Axiom. The sum of the angles of a linear pair is $180^\circ$.

This is the geometric form of the statement "a straight angle measures $180^\circ$". You can take it as an axiom or derive it from the definition of a straight angle as one of measure $180^\circ$.

A picture. Imagine a line $\overleftrightarrow{AOB}$ and a ray $\overrightarrow{OC}$ from a point $O$ on the line, with $C$ not on the line. Then $\angle AOC$ and $\angle COB$ are a linear pair, and $\angle AOC + \angle COB = 180^\circ.$

The converse is also true. If two adjacent angles sum to $180^\circ$ at a vertex with one shared arm, then the other two arms form a single straight line. This is often used to prove that three points are collinear.

Using the linear pair axiom

If you know one of the two angles in a linear pair, you know the other.

Example. If $\angle AOC = 65^\circ$, then $\angle COB = 180^\circ - 65^\circ = 115^\circ$.

You can also use a linear pair to set up an equation when an unknown is involved. If $\angle AOC = (2x + 10)^\circ$ and $\angle COB = (x + 50)^\circ$, then $(2x + 10) + (x + 50) = 180$, giving $3x = 120$, $x = 40$. The two angles are then $90^\circ$ and $90^\circ$ , they are a right-angle pair.

Worked examples

Example 1. $\angle AOC$ and $\angle BOC$ form a linear pair. If $\angle AOC = 75^\circ$, find $\angle BOC$.

$\angle BOC = 180^\circ - 75^\circ = 105^\circ$.

Example 2. Two angles of a linear pair are in the ratio $2 : 3$. Find them.

Let the angles be $2x$ and $3x$. Their sum is $180^\circ$: $5x = 180 \Rightarrow x = 36$. Angles: $72^\circ$ and $108^\circ$.

Example 3. $\angle ABD = 70^\circ$ and $\angle DBC = 110^\circ$. Is $A, B, C$ collinear?

The two angles share the arm $BD$ and add to $180^\circ$. By the converse of the linear pair axiom, $A, B, C$ are collinear.

Example 4. Find the complement of $35^\circ$ and the supplement of $35^\circ$.

Complement: $90 - 35 = 55^\circ$. Supplement: $180 - 35 = 145^\circ$.

Example 5. Two adjacent angles are complementary and one is twice the other. Find them.

Let the angles be $x$ and $2x$ with $x + 2x = 90$: $3x = 90, x = 30$. Angles: $30^\circ$ and $60^\circ$.

Try it yourself

1. Define linear pair in your own words.
2. The angles of a linear pair are $(3x + 5)^\circ$ and $(2x + 15)^\circ$. Find $x$ and the two angles.
3. Find the supplement of $122^\circ$.
4. Find the complement of $48^\circ$.
5. Two angles are supplementary and one is $20^\circ$ less than the other. Find both.
6. State the converse of the linear pair axiom and use it: $\angle ABD = 105^\circ$ and $\angle DBC = 75^\circ$, are $A, B, C$ collinear?
7. An angle is one-fourth of its supplement. Find the angle.
8. Sketch two adjacent angles of measure $50^\circ$ and $130^\circ$. Are they a linear pair?
9. If $\angle AOB = 120^\circ$ and $\overrightarrow{OC}$ bisects $\angle AOB$, find each half.
10. Define complementary and supplementary angles.

Pitfalls / Insight

• Adjacent does not imply linear pair. The non-common arms must form a straight line.
• Complementary $\ne$ supplementary. Complementary $= 90^\circ$, supplementary $= 180^\circ$.
• Use the converse. When you must prove three points are collinear, sum two adjacent angles and check for $180^\circ$.

Insight. The linear pair axiom is a workhorse. Every time you see two angles with a shared arm on a straight line, the rest of the geometry often falls out from "they sum to $180^\circ$". Memorise this fact and look for it everywhere.