Lesson 4: Angles in a Triangle

Property

When two lines intersect at a point, four angles are formed.
Angles on a straight line are supplementary, and the sum of their measures is $180^\circ$.
Angles opposing each other at a vertex are called vertical angles, and they are equal in measure.

Examples

• Two lines intersect. One angle is $40^\circ$. The angle opposite (vertical) to it is also $40^\circ$. The angles adjacent (supplementary) to it are each $180^\circ - 40^\circ = 140^\circ$.
• In an intersection, an angle $\angle A$ and an angle $\angle B$ are supplementary. If the measure of $\angle A$ is $110^\circ$, then the measure of $\angle B$ is $180^\circ - 110^\circ = 70^\circ$.
• Two intersecting lines form four angles. If one angle is a right angle ($90^\circ$), its vertical angle is also $90^\circ$, and its supplementary angles are also $180^\circ - 90^\circ = 90^\circ$. All four angles are right angles.

Explanation

Think of an 'X'. Angles side-by-side on a straight line are 'supplements' that complete a half-circle ($180^\circ$). Angles across from each other are 'vertical' and are perfect mirror images, so they must be equal.

Property

If two lines are parallel, and a third line (a transversal) cuts across both, then corresponding angles at the points of intersection have the same measure. The translation that moves the intersection point on the first line to the intersection point on the second line will also map the first line onto the second, showing the angles are congruent.

Conversely, if a transversal cuts across two lines such that the corresponding angles are equal, then the two lines are parallel.

Examples

• Lines $m$ and $n$ are parallel and cut by transversal $t$. If an upper-left angle at the first intersection is $125^\circ$, the upper-left angle at the second intersection is also $125^\circ$.