Math Core

In the foundations unit you learned that vertical angles are congruent and that a linear pair adds to $180^\circ$ . You probably believed it because the pictures looked right. Now you have the tools to prove these facts, and once a fact is proven it becomes a theorem you can use as a reason in every proof that follows.

The starting point: the Linear Pair Postulate

Recall that a linear pair is two adjacent angles whose non-shared sides are opposite rays, so together they form a straight line. We accept one fact about them without proof.

Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.

We also need the definitions of supplementary (measures add to $180^\circ$ ), complementary (measures add to $90^\circ$ ) and right angle (measure $90^\circ$ ). Everything in this lesson is built from those.

Right angles are congruent

Right Angles Congruence Theorem: All right angles are congruent.

Proof. If $\angle A$ and $\angle B$ are right angles, then $m\angle A = 90^\circ$ and $m\angle B = 90^\circ$ by the definition of a right angle. By the Transitive (or Substitution) Property of Equality, $m\angle A = m\angle B$ , so $\angle A \cong \angle B$ by the definition of congruent angles.

Congruent supplements and complements

Suppose $\angle 1$ and $\angle 2$ are both supplementary to $\angle 3$ . If $m\angle 3 = 70^\circ$ , then both $\angle 1$ and $\angle 2$ must measure $110^\circ$ . That works for any measure of $\angle 3$ .

Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

Congruent Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

Worked example: Proving the Congruent Supplements Theorem

Given: $\angle 1$ and $\angle 3$ are supplementary; $\angle 2$ and $\angle 3$ are supplementary. Prove: $\angle 1 \cong \angle 2$ .

#	statement	reason
1	$\angle 1$ and $\angle 3$ are supplementary; $\angle 2$ and $\angle 3$ are supplementary	Given
2	$m\angle 1 + m\angle 3 = 180^\circ$ ; $m\angle 2 + m\angle 3 = 180^\circ$	Definition of supplementary angles
3	$m\angle 1 + m\angle 3 = m\angle 2 + m\angle 3$	Substitution (or Transitive) Property of Equality
4	$m\angle 1 = m\angle 2$	Subtraction Property of Equality
5	$\angle 1 \cong \angle 2$	Definition of congruent angles

Vertical angles

When two lines cross, they form two pairs of vertical angles: angles opposite each other, sharing only the vertex.

Two intersecting lines form four angles. Angles 1 and 3 are vertical, and so are angles 2 and 4.

In the diagram, $\angle 1$ and $\angle 3$ are vertical, and so are $\angle 2$ and $\angle 4$ . Neighboring angles, like $\angle 1$ and $\angle 2$ , form linear pairs.

Worked example: Proving the Vertical Angles Theorem

Given: $\angle 1$ and $\angle 3$ are vertical angles (as in the diagram). Prove: $\angle 1 \cong \angle 3$ .

The idea: both $\angle 1$ and $\angle 3$ form a linear pair with $\angle 2$ , so both are supplementary to $\angle 2$ .

#	statement	reason
1	$\angle 1$ and $\angle 3$ are vertical angles	Given
2	$\angle 1$ and $\angle 2$ form a linear pair; $\angle 2$ and $\angle 3$ form a linear pair	Definition of linear pair (from the diagram)
3	$\angle 1$ and $\angle 2$ are supplementary; $\angle 2$ and $\angle 3$ are supplementary	Linear Pair Postulate
4	$\angle 1 \cong \angle 3$	Congruent Supplements Theorem

Short and clean, because the Congruent Supplements Theorem does the algebra for you.

Angle theorems you can now cite

Linear Pair Postulate: a linear pair is supplementary.
Right Angles Congruence Theorem: all right angles are congruent.
Congruent Supplements Theorem: supplements of the same (or congruent) angles are congruent.
Congruent Complements Theorem: complements of the same (or congruent) angles are congruent.
Vertical Angles Theorem: vertical angles are congruent.

Common mistake

Diagrams can show that angles are adjacent or form a linear pair, but not that they're congruent or right. You may read collinearity, betweenness and adjacency from a figure. You may not assume a right angle or equal measures unless they're marked or given. Also, vertical angles are congruent, not supplementary (unless both happen to be right angles).

Using the theorems to find measures

Worked example: Finding all four angles

Two lines intersect, and $m\angle 1 = 4x + 10$ and $m\angle 3 = 6x - 30$ , where $\angle 1$ and $\angle 3$ are vertical. Find all four angle measures.

Solution. By the Vertical Angles Theorem, $4x + 10 = 6x - 30$ , so $40 = 2x$ and $x = 20$ . Then $m\angle 1 = m\angle 3 = 4(20) + 10 = 90^\circ$ . Here $\angle 2$ forms a linear pair with $\angle 1$ , so $m\angle 2 = 180^\circ - 90^\circ = 90^\circ$ , and $m\angle 4 = 90^\circ$ too. The lines are perpendicular.

Worked example: A linear pair equation

$\angle ABD$ and $\angle DBC$ form a linear pair, with $m\angle ABD = 3x + 15$ and $m\angle DBC = 2x - 10$ . Find both measures.

Solution. By the Linear Pair Postulate, $(3x + 15) + (2x - 10) = 180$ . So $5x + 5 = 180$ , $5x = 175$ and $x = 35$ . Then $m\angle ABD = 3(35) + 15 = 120^\circ$ and $m\angle DBC = 2(35) - 10 = 60^\circ$ . Check: $120 + 60 = 180$ .

Tip

Decide the relationship before writing the equation. Vertical angles: set the expressions equal. Linear pair: make them add to $180$ . Complementary: add to $90$ .

Practice

Practice 1

Two lines intersect. One of the four angles measures $62^\circ$ . What is the measure, in degrees, of an angle that forms a linear pair with it?