Vertical Angles, Right Angles and Perpendicular Lines | Geometry | LevelKite

You have already used vertical angles to find angle measures. In this lesson, you will prove why they always match, and then you will use that result with right angles and perpendicular lines.

Here are the terms you need. Opposite rays start at the same point and go in opposite directions along one line. Adjacent angles share a vertex and one side, with no overlap. A linear pair is two adjacent angles whose other sides are opposite rays. Vertical angles are two angles formed by two intersecting lines that share only their vertex, so the sides of one are opposite rays to the sides of the other.

Lines

AB

and

CD

intersect at

E

.

\angle AEC

and

\angle BED

are vertical angles, and

\angle AED

forms a linear pair with each of them.

Throughout this lesson, $E$ is between $A$ and $B$ and between $C$ and $D$ . So rays $EA$ and $EB$ are opposite rays, and so are rays $EC$ and $ED$ . You know this because it is stated here, not because of how a picture looks. The figures are not drawn to scale.

Try a number first. Write $m\angle AED$ for the measure of $\angle AED$ , and suppose $m\angle AED = 130^\circ$ in the figure above. The Supplement Theorem says a linear pair is supplementary, which means the two measures add to $180^\circ$ . So $m\angle AEC = 180^\circ - 130^\circ = 50^\circ$ and $m\angle BED = 180^\circ - 130^\circ = 50^\circ$ . Each value is greater than $0^\circ$ and at most $180^\circ$ , as an angle measure must be.

The vertical angles match because each one is what is left when you take the same angle away from $180^\circ$ . That idea does not depend on the number $130$ , so it can become a proof.

Key concept

Vertical Angles Theorem

If two angles are vertical angles, then they are congruent. Congruent angles are angles with equal measures. The condition matters: the two angles must be formed by two intersecting lines and share only their vertex. For vertical angles $\angle AEC$ and $\angle BED$ , the theorem says:

\angle AEC \cong \angle BED

Worked example

Plan the proof of the Vertical Angles Theorem

$\angle AEC$ and $\angle BED$ are vertical angles formed by lines $AB$ and $CD$ intersecting at $E$ . Plan a proof that $\angle AEC \cong \angle BED$ . The plan is to show that both angles are supplementary to $\angle AED$ , and then compare their measures.

Find the linear pairs. $\angle AEC$ and $\angle AED$ share ray $EA$ , and rays $EC$ and $ED$ are opposite rays. $\angle BED$ and $\angle AED$ share ray $ED$ , and rays $EB$ and $EA$ are opposite rays. Reason: Definition of linear pair.
$\angle AEC \text{ with } \angle AED, \text{ and } \angle BED \text{ with } \angle AED, \text{ are linear pairs}$
Each linear pair is supplementary. Reason: Supplement Theorem.
$\angle AEC \text{ with } \angle AED, \text{ and } \angle BED \text{ with } \angle AED, \text{ are supplementary}$
Write each fact as an equation, so you can work with the measures. Reason: Definition of supplementary angles.
$m\angle AEC + m\angle AED = 180^\circ,\quad m\angle BED + m\angle AED = 180^\circ$
Both sums equal $180^\circ$ , so replace $180^\circ$ in the first equation with the sum from the second equation. Reason: Substitution Property of Equality.
$m\angle AEC + m\angle AED = m\angle BED + m\angle AED$
Subtract $m\angle AED$ from both sides. This is the same take-away idea you saw with $130^\circ$ . Reason: Subtraction Property of Equality.
$m\angle AEC = m\angle BED$
Equal measures mean the angles are congruent. Reason: Definition of congruent angles.
$\angle AEC \cong \angle BED$

Answer

The plan reaches $\angle AEC \cong \angle BED$ , because both angles are supplementary to the same angle, $\angle AED$ .

A two-column proof writes that plan as a table: each row pairs a statement with the reason it is true. Every reason is the word Given or the exact name of a definition, postulate, property or proved theorem. A reason is never how a figure looks.

Given: $\angle AEC$ and $\angle BED$ are vertical angles formed by lines $AB$ and $CD$ intersecting at $E$ .

Prove: $\angle AEC \cong \angle BED$

The Given is what you may start from, and the Prove is what you must reach. The letters can name any two intersecting lines, so this proof covers every pair of vertical angles.

Proof of the Vertical Angles Theorem

Statement	Reason
$\angle AEC$ and $\angle BED$ are vertical angles formed by lines $AB$ and $CD$ intersecting at $E$ .	Given
$\angle AEC$ and $\angle AED$ form a linear pair.	Definition of linear pair
$\angle BED$ and $\angle AED$ form a linear pair.	Definition of linear pair
$\angle AEC$ and $\angle AED$ are supplementary.	Supplement Theorem
$\angle BED$ and $\angle AED$ are supplementary.	Supplement Theorem
$m\angle AEC + m\angle AED = 180^\circ$	Definition of supplementary angles
$m\angle BED + m\angle AED = 180^\circ$	Definition of supplementary angles
$m\angle AEC + m\angle AED = m\angle BED + m\angle AED$	Substitution Property of Equality
$m\angle AEC = m\angle BED$	Subtraction Property of Equality
$\angle AEC \cong \angle BED$	Definition of congruent angles

Worked example

Use the Vertical Angles Theorem to find measures

$\angle AEC$ and $\angle BED$ are vertical angles formed by lines $AB$ and $CD$ intersecting at $E$ . $m\angle AEC = (3x + 10)^\circ$ and $m\angle BED = (5x - 30)^\circ$ . Find $x$ and both measures, then find $m\angle AED$ .

Use the theorem in its stated direction: the angles are vertical, so the Vertical Angles Theorem makes them congruent. The Definition of congruent angles then makes their measures equal.
$3x + 10 = 5x - 30$
Subtract $3x$ from both sides (Subtraction Property of Equality).
$10 = 2x - 30$
Add $30$ to both sides (Addition Property of Equality).
$40 = 2x$
Divide both sides by $2$ (Division Property of Equality).
$x = 20$
Substitute $x = 20$ into both expressions. Check: the measures are equal, and each is greater than $0^\circ$ and at most $180^\circ$ , so you keep $x = 20$ .
$m\angle AEC = 3(20) + 10 = 70^\circ,\quad m\angle BED = 5(20) - 30 = 70^\circ$
$\angle AEC$ and $\angle AED$ form a linear pair, so they are supplementary (Supplement Theorem) and their measures add to $180^\circ$ (Definition of supplementary angles). Subtract $70^\circ$ from $180^\circ$ . Check: $110^\circ$ is greater than $0^\circ$ and at most $180^\circ$ .
$m\angle AED = 180^\circ - 70^\circ = 110^\circ$

Answer

The value of $x$ is $20$ , so $\angle AEC$ and $\angle BED$ each measure $70^\circ$ , and $\angle AED$ measures $110^\circ$ .

Now turn to right angles. A right angle is an angle that measures exactly $90^\circ$ . Perpendicular lines are two lines that intersect to form a right angle. You write $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$ .

This definition works in both directions. If two lines are perpendicular, they form a right angle. If two intersecting lines form a right angle, they are perpendicular.

The table below gives three theorems that build on these definitions. Each one has its own condition, so check that the condition is met before you use the theorem.

Three theorems about right angles and perpendicular lines

Theorem	What it says	Why it is true
Right Angle Congruence Theorem	All right angles are congruent.	Each measures $90^\circ$ (Definition of right angle), so they are congruent (Definition of congruent angles).
Four Right Angles Theorem	Perpendicular lines intersect to form four right angles.	The proof below uses the Supplement Theorem and the Vertical Angles Theorem.
Congruent Linear Pair Theorem	If two congruent angles form a linear pair, then they are right angles.	Equal measures that add to $180^\circ$ (Supplement Theorem) are each $90^\circ$ (Division Property of Equality).

The square at

E

between rays

EC

and

EB

states that

\angle CEB

is a right angle, so lines

AB

and

CD

are perpendicular.

In the figure above, the square at $E$ between rays $EC$ and $EB$ states that $\angle CEB$ is a right angle. That mark is a given. An angle that only looks square tells you nothing.

The definition of perpendicular lines promises only one right angle, so label the points to make it $\angle CEB$ . The Supplement Theorem and the Vertical Angles Theorem give you the other three, so the theorem you just proved now works as a reason.

Given: $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$ at $E$ , and $\angle CEB$ is a right angle.

Prove: $\angle AEC$ , $\angle CEB$ , $\angle BED$ and $\angle AED$ are right angles.

Proof of the Four Right Angles Theorem

Statement	Reason
$\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$ at $E$ , and $\angle CEB$ is a right angle.	Given
$m\angle CEB = 90^\circ$	Definition of right angle
$\angle AEC$ and $\angle CEB$ form a linear pair.	Definition of linear pair
$\angle AEC$ and $\angle CEB$ are supplementary.	Supplement Theorem
$m\angle AEC + m\angle CEB = 180^\circ$	Definition of supplementary angles
$m\angle AEC + 90^\circ = 180^\circ$	Substitution Property of Equality
$m\angle AEC = 90^\circ$	Subtraction Property of Equality
$\angle AED \cong \angle CEB$ and $\angle BED \cong \angle AEC$	Vertical Angles Theorem
$m\angle AED = m\angle CEB$ and $m\angle BED = m\angle AEC$	Definition of congruent angles
$m\angle AED = 90^\circ$ and $m\angle BED = 90^\circ$	Substitution Property of Equality
$\angle AEC$ , $\angle CEB$ , $\angle BED$ and $\angle AED$ are right angles.	Definition of right angle

Lines

AB

and

CD

are perpendicular at

E

, as the square between rays

EC

and

EB

states. Ray

EF

lies inside

\angle CEB

, forming

\angle CEF

and

\angle FEB

.

Worked example

Use perpendicular lines to find measures

In the figure above, $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$ at $E$ , and $F$ lies in the interior of $\angle CEB$ . $\angle CEF$ measures $(3x + 2)^\circ$ and $\angle FEB$ measures $(2x + 8)^\circ$ . Find $x$ and both measures.

The lines are perpendicular, so the Four Right Angles Theorem says all four angles at $E$ are right angles, including $\angle CEB$ . This runs from perpendicular lines to right angles. The Definition of right angle gives the measure.
$m\angle CEB = 90^\circ$
$F$ is in the interior of $\angle CEB$ , so use the Angle Addition Postulate.
$m\angle CEF + m\angle FEB = m\angle CEB$
Substitute the given expressions and $90$ (Substitution Property of Equality).
$(3x + 2) + (2x + 8) = 90$
Combine like terms on the left side.
$5x + 10 = 90$
Subtract $10$ from both sides (Subtraction Property of Equality), then divide both sides by $5$ (Division Property of Equality).
$5x = 80,\quad x = 16$
Substitute $x = 16$ . Check: each measure is greater than $0^\circ$ and at most $180^\circ$ , and $50^\circ + 40^\circ = 90^\circ$ , so you keep $x = 16$ .
$m\angle CEF = 3(16) + 2 = 50^\circ,\quad m\angle FEB = 2(16) + 8 = 40^\circ$

Answer

The value of $x$ is $16$ , so $\angle CEF$ measures $50^\circ$ and $\angle FEB$ measures $40^\circ$ .

Worked example

Show that two lines are perpendicular

Lines $AB$ and $CD$ intersect at $E$ , and $\angle AEC \cong \angle CEB$ . $m\angle AEC = (x^2 - 10)^\circ$ and $m\angle CEB = (9x)^\circ$ . Find $x$ and both measures, and decide whether the lines are perpendicular.

$\angle AEC$ and $\angle CEB$ form a linear pair (Definition of linear pair), and they are given as congruent. Both conditions hold, so the Congruent Linear Pair Theorem makes them right angles. The Definition of right angle gives their measures.
$m\angle AEC = 90^\circ,\quad m\angle CEB = 90^\circ$
Substitute the expression for $m\angle AEC$ (Substitution Property of Equality).
$x^2 - 10 = 90$
Add $10$ to both sides (Addition Property of Equality).
$x^2 = 100$
Take the square root of both sides. Both $10$ and $-10$ have a square of $100$ .
$x = 10 \text{ or } x = -10$
Test $x = -10$ in the other angle. A measure must be greater than $0^\circ$ , so you reject $x = -10$ .
$m\angle CEB = 9(-10) = -90^\circ$
Test $x = 10$ . Both measures are $90^\circ$ , which is greater than $0^\circ$ and at most $180^\circ$ , so you keep $x = 10$ .
$m\angle AEC = 10^2 - 10 = 90^\circ,\quad m\angle CEB = 9(10) = 90^\circ$
Use the definition in the direction from right angle to perpendicular lines: two intersecting lines that form a right angle are perpendicular. Reason: Definition of perpendicular lines.
$\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$

Answer

The value of $x$ is $10$ , $\angle AEC$ and $\angle CEB$ each measure $90^\circ$ , and line $AB$ is perpendicular to line $CD$ .

The takeaway: Vertical angles are congruent because each one is supplementary to the same angle. With the Supplement Theorem, that result turns one right angle at an intersection into four. A congruent linear pair also gives right angles, which shows that two lines are perpendicular. Each time you use one of these theorems, check its condition and name it exactly. Next, you will write your own two-column proof about angles.