SAT Math: Lines, angles, and triangles

Lines, angles and triangles, including similar triangles.

Geometry and Trigonometry · Lines, angles, and triangles
Both modules
13 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise lines, angles, and triangles →Adaptive SAT module

What the test covers

Parallel-line angles, triangle sums, similar triangles

Key ideas

Angles on a line add to $180^{\circ}$ ; angles in a triangle add to $180^{\circ}$ ; corresponding and alternate angles on parallel lines are equal.
Similar triangles: corresponding sides are in the same ratio $k$ , and areas are in the ratio $k^{2}$ .
Angles in a ratio $p : q : s$ are $\frac{p}{p + q + s} \times 180^{\circ}$ , and so on.

Common mistakes

Match corresponding sides by position, not by length.
Area scales by $k^{2}$ , not $k$ .

Do it in Desmos

Solve angle equations like any linear equation (Desmos intersection if needed).

Worked example

The measures of the angles of a triangle are in the ratio $3 : 4 : 5$ . What is the measure, in degrees, of the largest angle?

$105$
$60$
$50$
$75$

Answer: D: $75$

$3 + 4 + 5 = 12$ parts share $180^{\circ}$ : $15^{\circ}$ per part. Largest $= 5 \times 15 = 75^{\circ}$ .

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

The measures of the angles of a triangle are in the ratio $2 : 5 : 8$ . What is the measure, in degrees, of the largest angle?

$84$
$60$
$80$
$96$

Show the answer and solution

Answer: D: $96$

$2 + 5 + 8 = 15$ parts share $180^{\circ}$ : $12^{\circ}$ per part. Largest $= 8 \times 12 = 96^{\circ}$ .

Question 2

Triangles $A B C$ and $D E F$ are similar, with side $A B = 6$ corresponding to side $D E = 15$ . The area of triangle $A B C$ is 20. What is the area of triangle $D E F$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 125

The scale factor is $\frac{15}{6} = 2.5$ , so areas scale by ${2.5}^{2} = 6.25$ : $20 \times 6.25 = 125$ .

Question 3

Triangles $P Q R$ and $S T U$ are similar, with side $P Q = 2$ corresponding to side $S T = 7$ . The area of triangle $P Q R$ is 12. What is the area of triangle $S T U$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 147

The scale factor is $\frac{7}{2}$ , so areas scale by ${(\frac{7}{2})}^{2}$ : $12 \times \frac{49}{4} = 147$ .

Question 4

The measures of the angles of a triangle are in the ratio $2 : 3 : 7$ . What is the measure, in degrees, of the largest angle?

$120$
$90$
$75$
$105$