SAT lines, angles and triangles: the facts you need

The key facts

Straight line: angles that form a line add to $180^\circ$ .
Vertical angles: the opposite angles where two lines cross are equal.
Parallel lines and a transversal: corresponding angles are equal, alternate interior angles are equal, and same-side interior angles add to $180^\circ$ .
Triangle sum: the three angles add to $180^\circ$ . An exterior angle equals the sum of the two far interior angles.
Isosceles triangle: two equal sides means the two angles opposite them are equal.
Similar triangles: same angles, sides in the same ratio. Two pairs of equal angles is enough to prove it.

\triangle ABC \sim \triangle DEF \quad\Longrightarrow\quad \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}

A line drawn parallel to one side of a triangle cuts off a smaller triangle that is similar to the whole one. The SAT uses this setup often.

Worked examples

Example 1: parallel lines

Problem Two parallel lines are cut by a transversal. One angle measures $(3x + 10)^\circ$ , and its corresponding angle measures $(5x - 30)^\circ$ . What is the measure of each angle?

Corresponding angles are equal.
$3x + 10 = 5x - 30$
Subtract $3x$ and add 30.
$40 = 2x$
Divide by 2.
$x = 20$
Find the angle.
$3(20) + 10 = 70$

Answer Each angle measures $70^\circ$ .

Example 2: an exterior angle

Problem In triangle ABC, angle A measures $48^\circ$ . The exterior angle at C measures $115^\circ$ . What is the measure of angle B?

An exterior angle equals the two far interior angles added together.
$48 + B = 115$
Subtract 48.
$B = 67$

Answer $67^\circ$

Example 3: similar triangles

Problem Triangle PQR is similar to triangle STU, with P matching S, Q matching T and R matching U. If $PQ = 6$ , $ST = 15$ and $QR = 8$ , what is TU?

Find the scale factor from the matching sides PQ and ST.
$\frac{15}{6} = 2.5$
TU matches QR, so multiply by the same factor.
$TU = 8 \times 2.5 = 20$

Answer $TU = 20$

Example 4 (SAT-hard): a parallel line inside a triangle

Problem In triangle ABC, point D is on side AB and point E is on side AC so that DE is parallel to BC. If $AD = 4$ , $DB = 6$ and $DE = 5$ , what is BC?

DE parallel to BC makes triangle ADE similar to triangle ABC (they share angle A, and the parallel lines make equal corresponding angles).
Use the whole side AB, not DB.
$AB = 4 + 6 = 10$
Set up the proportion: small over big for both pairs.
$\frac{AD}{AB} = \frac{DE}{BC}$
Substitute and solve.
$\frac{4}{10} = \frac{5}{BC}$

Answer $BC = 12.5$

Common mistakes

Using DB instead of AB in the proportion. The small triangle's side AD matches the big triangle's side AB, which is $4 + 6$ . Fix: draw the two triangles separately.
Setting same-side interior angles equal. They add to $180^\circ$ . Fix: equal angles look the same size in the figure; supplementary ones look different.
Matching sides in the wrong order. In $\triangle PQR \sim \triangle STU$ , QR matches TU because the letters are in the same positions. Fix: use the order of the letters.
Stopping at x. If the question asks for the angle, plug x back in. Fix: reread the question.
Trusting how a figure looks. College Board's directions say figures are drawn to scale unless noted otherwise, but use the given facts, not a guess from the picture.

Quick method

Practice

5 SAT-style questions

Two angles form a straight line. One measures $(2x + 15)^\circ$ and the other $(4x - 15)^\circ$ . What is x?
1. 25
2. 30
3. 45
4. 60
Show answer
Answer: 30
Angles on a line add to 180: $6x = 180$ , so $x = 30$ . Setting the angles equal instead gives 15, and 45 is the first angle's size when $x = 15$ .
The angles of a triangle are in the ratio $2 : 3 : 7$ . What is the measure of the largest angle?
1. $70^\circ$
2. $84^\circ$
3. $105^\circ$
4. $120^\circ$
Show answer
Answer: $105^\circ$
There are $2 + 3 + 7 = 12$ parts, so one part is $180 \div 12 = 15$ . The largest angle is $7 \times 15 = 105$ .
An isosceles triangle has a vertex angle of $40^\circ$ . What is the measure of each base angle?
1. $40^\circ$
2. $50^\circ$
3. $70^\circ$
4. $140^\circ$
Show answer
Answer: $70^\circ$
The two base angles are equal and share $180 - 40 = 140$ degrees, so each is 70. 140 forgets to split between the two angles.
A 6-foot person casts a 4-foot shadow at the same time a tree casts a 30-foot shadow. How tall is the tree?
1. 20 feet
2. 36 feet
3. 45 feet
4. 180 feet
Show answer
Answer: 45 feet
The sun makes similar triangles: $\frac{6}{4} = \frac{h}{30}$ , so $h = 45$ . 20 flips the ratio to shadow over height.
Student-produced response: in triangle ABC, DE is parallel to BC, with D on AB and E on AC. If $AD = 3$ , $AB = 12$ and $BC = 20$ , what is DE?
Show answer
Answer: 5
Triangle ADE is similar to triangle ABC with scale factor $\frac{3}{12} = \frac{1}{4}$ . So $DE = \frac{1}{4} \times 20 = 5$ .

Frequently asked questions

How do I know two triangles are similar?

Show two pairs of equal angles (AA). On the SAT the usual clues are a shared angle plus parallel lines, or two right angles plus a shared angle. Once they are similar, every pair of matching sides has the same ratio.

What is the exterior angle theorem?

An exterior angle of a triangle equals the sum of the two interior angles that are not next to it. It comes from the triangle sum: both the exterior angle and those two angles add with the third angle to $180^\circ$ .

Are SAT figures drawn to scale?

College Board's math directions say figures provided are drawn to scale unless otherwise indicated. Still, solve with the given numbers and facts. Use the picture to check that your answer looks reasonable.

Sources

College Board: SAT Math, Geometry and Trigonometry skills, accessed October 1, 2026
College Board: SAT Bluebook test directions (math directions), accessed October 1, 2026

Lines, angles and triangles on the SAT

The key facts

Worked examples

Common mistakes

Quick method

Practice

Frequently asked questions

Sources

Try asking Ducky

The key facts

Worked examples

Common mistakes

Quick method

Practice

Frequently asked questions

Sources

Related

Try asking Ducky