Use line and angle relationships

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Difficulty: Beginner
Estimated time: 28 minutes
Domains: Geometry and Trigonometry
Techniques: Vertical-anglesLinear-pairsParallel-linesTransversalsAngle-sums

What you’ll learn

Spot vertical angles and linear pairs where two lines cross.
Use corresponding, alternate, and same-side angles when a transversal crosses parallel lines.
Use the totals $90^\circ$ , $180^\circ$ , and $360^\circ$ .
Turn a diagram’s markings into an equation.
Answer what the question actually asks for, whether that’s an angle or a variable.

Prerequisites

Solve linear equations0%
Use equivalent steps to solve the angle equation after the geometry relationship is identified.

Why this matters on the SAT

Turn a crowded diagram into one equation

An SAT diagram can pack in several lines, angle expressions, and labels. But the work usually starts with one question: are the marked angles equal, or do they add up to a known total? Once you know which, the picture turns into a short equation. Here’s a typical example.

Solution to the example

The two labeled angles sit opposite each other where the lines cross, so they’re vertical angles. Vertical angles are equal:

3x+7=76.

Now solve the linear equation:

\begin{aligned} 3x&=69\\[1.4em] x&=23. \end{aligned}

The answer is B.

Choice C is the trap. It comes from adding the two angles to get $180^\circ$ , which is what you’d do if they sat side by side. They sit opposite each other, so they’re equal instead.

SAT example

The labeled angles are opposite angles at the same intersection. Figure not drawn to scale.

Two straight lines intersect at point $P$ . The measures of two angles are shown. What is the value of $x$ ?

A
$21$
B
$23$
C
$\frac{97}{3}$
D
$69$

Read the markings before the shape

Don’t judge an angle by how wide it looks. SAT figures aren’t always drawn to scale, so trust what’s stated and marked instead.

A straight path through a point is a line. The two rays pointing opposite ways along it form a $180^\circ$ straight angle.
Matching arrow marks on two lines mean they’re parallel. A statement like $m\parallel n$ tells you the same thing.
A small square in a corner marks a $90^\circ$ right angle.
A transversal is a line that crosses two or more other lines.

If two lines only look parallel, with no arrow marks and no statement, you can’t use parallel-line facts. The picture shows you where to look, but only what’s given lets you write the equation.

Angles inside triangles come next, in Use triangle angle theorems.

Check your understanding:

Two lines in a diagram look parallel, but there are no arrow marks and the text never says they’re parallel. Can you say that corresponding angles are equal?

Use the two angle families at an intersection

When two straight lines cross, they make four angles around the point.

Vertical angles sit opposite each other. They’re equal.
A linear pair is two angles side by side whose outer rays form a straight line. They add up to $180^\circ$ .

In the figure, the two $a^\circ$ angles are vertical, and so are the two $b^\circ$ angles. Any $a^\circ$ angle and a $b^\circ$ angle next to it form a linear pair, so

a+b=180.

That means one angle gives you all four. The angle opposite it has the same measure, and each angle next to it is $180^\circ$ minus that measure.

Opposite angles are equal. Side-by-side angles form a linear pair and add to

180^\circ

Check your understanding:

Two lines cross, and one of the angles measures $128^\circ$ . What are the angle opposite it and the angle next to it?

Common mistake:

If you’ve set two touching angles equal, check where they sit. Vertical angles don’t share a side; they’re across from each other. Two touching angles where lines cross usually form a linear pair, so they add to $180^\circ$ instead. As a check, an obtuse angle should pair with an acute one.

Carry angles across parallel lines

Here’s what makes parallel lines so useful: whatever angles a transversal makes where it crosses one line, it makes again where it crosses the other.

So you don’t need a separate picture for every position. At one crossing, vertical angles match and side-by-side angles add to $180^\circ$ , as you just saw. The parallel lines copy that pattern to the second crossing. The small angles all match, the big angles all match, and a small one plus a big one makes $180^\circ$ . In the figure, every $a^\circ$ angle is equal, and

a+b=180.

The names tell you which pair you’re looking at. With parallel lines, these three pairs are equal:

Corresponding angles sit in the same corner at the two crossings.
Alternate interior angles sit between the parallel lines, on opposite sides of the transversal.
Alternate exterior angles sit outside the parallel lines, on opposite sides of the transversal.

This pair adds up instead:

Same-side interior angles sit between the parallel lines, on the same side of the transversal. They add to $180^\circ$ .

The same two angles,

a^\circ

and

b^\circ

, show up again at the second crossing.

Check your understanding:

In the figure, say $a=64$ . What is $b$ , and which relationship tells you?

Match the pieces to the right total

Sometimes a diagram splits a familiar angle into several pieces that don’t overlap. Add every piece exactly once, and match the sum to what the pieces fill:

pieces that fill a right angle add to $90^\circ$ ;
pieces that fill a straight angle add to $180^\circ$ ;
pieces that go all the way around a point add to $360^\circ$ .

The tricky part is counting each piece exactly once. So before you write the equation, trace it: start at one ray and move around the point, piece by piece, until you reach the last ray. That way you won’t skip a piece or count one twice.

Try it yourself:

Sketch three rays that split a straight angle into $x^\circ$ , $(2x+15)^\circ$ , and $45^\circ$ . Don’t solve it yet. Trace the straight angle from one side to the other and write its equation, checking that each piece shows up once.

Here’s the equation:

x+(2x+15)+45=180.

The total is $180^\circ$ because the two outer rays point in opposite directions. If the pieces kept going all the way around their shared endpoint, the total would be $360^\circ$ instead.

Example: Chain parallel and vertical relationships

Worked example

Use the parallel-line relationship to find

x

, then return to the vertical angle

y

. Figure not drawn to scale.

In the figure, $m\parallel n$ . A transversal intersects both lines. The labeled angles are $y^\circ$ , $(5x+22)^\circ$ , and $(3x+22)^\circ$ . What is the value of $y$ ?

Step 1

Name the first relationship

The angles $(5x+22)^\circ$ and $(3x+22)^\circ$ sit between the parallel lines, on the same side of the transversal. That makes them same-side interior angles, so they add to $180^\circ$ :

(5x+22)+(3x+22)=180.

This works only because the arrow marks tell you $m\parallel n$ . Without parallel lines, you couldn’t count on that $180^\circ$ .

Step 2

Solve for x

Combine like terms, then get $x$ by itself:

\begin{aligned} 5x+22+3x+22&=180\\[1.4em] 8x+44&=180\\[1.4em] 8x&=136\\[1.4em] x&=17. \end{aligned}

Step 3

Go back for y

Careful here: the question asks for $y$ , not $x$ . At the upper crossing, $y^\circ$ and $(5x+22)^\circ$ are vertical angles, so they’re equal:

\begin{aligned} y&=5x+22\\[1.4em] &=5(17)+22\\[1.4em] &=107. \end{aligned}

\boxed{y=107}.

A quick sense check: $107^\circ$ is obtuse, so the angle next to it should be acute. It is: it measures $180^\circ-107^\circ=73^\circ$ .

Check your understanding:

Keep the same positions, but make the two interior angles $(4z+8)^\circ$ and $(2z+40)^\circ$ . What is $z$ , and what’s the vertical angle opposite $(4z+8)^\circ$ ?

Common mistake:

Getting $x=17$ solves the equation, but not the question. Before you calculate, circle what the question asks for. Once you have $x$ , go back to the expression or angle relationship that gives you that target. As a last check, any ordinary angle in a diagram like this should measure between $0^\circ$ and $180^\circ$ .

Choose the relationship before the tool

On these questions, working by hand is usually quicker than Desmos:

Circle what the question asks for, like $y$ rather than $x$ .
Read what’s given, such as a statement like $m\parallel n$ or matching arrow marks.
Decide whether the angles are equal, like vertical angles, or add to $90^\circ$ , $180^\circ$ , or $360^\circ$ , like a linear pair.
Write one equation, solve it, and finish with the thing you circled.

Desmos can graph the two sides of a linear equation and show you its solution. What it can’t do is tell you which equation the diagram gives you:

3x+7=76

(3x+7)+76=180.

Only the diagram can tell you that. And once you’ve picked, the arithmetic here is short, so typing it into Desmos adds steps without saving any thinking. Work these by hand. If you like, use Desmos to check your algebra, but only after you’ve named the relationship.

Practice problems

For each one, start by asking: are these angles equal, or do they add up to a total?

Solve a linear pair

Practice problem

Two adjacent angles form a linear pair. Their measures are $(4x+12)^\circ$ and $(7x+3)^\circ$ . Which choice gives the value of $x$ ?

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Carry an angle across parallel lines

Practice problem

The labeled angles are alternate interior angles. Figure not drawn to scale.

In the figure, $r\parallel s$ . What is the measure, in degrees, of the angle labeled $(5x-4)^\circ$ ?

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Complete a full turn

Practice problem

Trace once around

P

and include each labeled region exactly once. Figure not drawn to scale.

The five labeled angles in the figure meet at point $P$ without overlapping and make one complete turn. What is the measure, in degrees, of the angle labeled $2x^\circ$ ?

Your answer

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Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Where two lines cross, vertical angles are equal and a linear pair adds to $180^\circ$ .
A transversal across parallel lines repeats the same two angles at each crossing: the small angles all match, the big angles all match, and a small one plus a big one makes $180^\circ$ .
Pieces that fill a right angle add to $90^\circ$ , a straight angle to $180^\circ$ , and a full turn around a point to $360^\circ$ .
Trust what’s stated and marked, not how the picture looks.
Name the relationship, write one equation, solve it, and answer what the question asks for.

Next lesson

Use triangle angle theorems

Extend angle-sum reasoning to triangle interiors, exteriors, and isosceles relationships.

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Practice

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329 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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