lines, angles, & triangles

The SAT loves to test your understanding of lines, angles, triangles, and the relationships between them. You’ll be asked to find unknown angles, use triangle sum, spot similarity, and apply proportional reasoning to geometric diagrams and word problems. Here’s everything you need for top scores—formulas, worked examples, and practice.

angles and lines

Angles measure rotation, in degrees (°). A straight line is $180^\circ$ , and a full circle is $360^\circ$ . Key types:

Acute: Less than $90^\circ$
Right: Exactly $90^\circ$
Obtuse: Between $90^\circ$ and $180^\circ$
Straight: Exactly $180^\circ$

For example:

x + 48^\circ = 90^\circ

If an angle is $48^\circ$ , its complement is $90^\circ - 48^\circ = 42^\circ$ .

triangle angle sum

In any triangle, the sum of the three interior angles is $180^\circ$ . If you know two, subtract their sum from $180^\circ$ to find the third.

For example:

\triangle ABC

\angle A = 60^\circ,\ \angle B = 85^\circ

\angle C = 180^\circ - 60^\circ - 85^\circ = 35^\circ

similar triangles & proportional reasoning

Two triangles are similar if their angles are the same (corresponding angles) and their sides are in proportion. The SAT tests you on using similar triangles to solve for unknown lengths and real-world applications (like shadows and ladders).

For example:

Triangle $ABC$ has sides 6, 8, 10. Triangle $DEF$ has sides 12, 16, 20. The triangles are similar (all sides are doubled). The ratios are $\frac{6}{12} = \frac{8}{16} = \frac{10}{20} = \frac{1}{2}$ .

right triangles & real-world setups

The SAT often asks you to set up and solve right triangles in context (shadows, ladders, buildings). Similar triangles can be used when two shapes share the same angles—proportions relate corresponding sides.

For example:

A 9-ft flagpole casts a 6-ft shadow. At the same time, a nearby tree casts a 10-ft shadow. How tall is the tree?
Set up a proportion:

\frac{9}{6} = \frac{x}{10}

6x = 90 \implies x = 15

The tree is 15 ft tall.

Test your skills with these SAT-style questions!

Question 1

In $\triangle XYZ$ , the measure of $\angle X$ is $24^\circ$ and the measure of $\angle Y$ is $98^\circ$ . What is the measure of $\angle Z$ ?

Question 2

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is $10$ feet tall and has a shadow that is $5$ feet long. At the same time, the shadow of the other tree is $2$ feet long. How tall, in feet, is the other tree?

Question 3

In triangles $LMN$ and $RST$ , angles $L$ and $R$ each have measure $60^\circ$ , $LN=10$ , and $RT=30$ . Which additional piece of information is sufficient to prove that triangle $LMN$ is similar to triangle $RST$ ?

✓

lines, angles, and triangles mastered

You now have the skills to solve SAT problems involving angle measures, triangle properties, and similarity setups!

Next Up

39: right triangles & trigonometry

master right triangles, the Pythagorean theorem, and basic trig

lines, angles, & triangles

angles and lines

triangle angle sum

similar triangles & proportional reasoning

right triangles & real-world setups

Question 1

In △XYZ\triangle XYZ△XYZ, the measure of ∠X\angle X∠X is 24∘24^\circ24∘ and the measure of ∠Y\angle Y∠Y is 98∘98^\circ98∘. What is the measure of ∠Z\angle Z∠Z?

Question 2

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 101010 feet tall and has a shadow that is 555 feet long. At the same time, the shadow of the other tree is 222 feet long. How tall, in feet, is the other tree?

Question 3

In triangles LMNLMNLMN and RSTRSTRST, angles LLL and RRR each have measure 60∘60^\circ60∘, LN=10LN=10LN=10, and RT=30RT=30RT=30. Which additional piece of information is sufficient to prove that triangle LMNLMNLMN is similar to triangle RSTRSTRST?

lines, angles, and triangles mastered

39: right triangles & trigonometry

In $\triangle XYZ$ , the measure of $\angle X$ is $24^\circ$ and the measure of $\angle Y$ is $98^\circ$ . What is the measure of $\angle Z$ ?

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is $10$ feet tall and has a shadow that is $5$ feet long. At the same time, the shadow of the other tree is $2$ feet long. How tall, in feet, is the other tree?

In triangles $LMN$ and $RST$ , angles $L$ and $R$ each have measure $60^\circ$ , $LN=10$ , and $RT=30$ . Which additional piece of information is sufficient to prove that triangle $LMN$ is similar to triangle $RST$ ?