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
- Calculator allowed (Desmos)
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 \(\tfrac{p}{p+q+s}\times180^\circ\), 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
Worked example
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?
- \(120\)
- \(90\)
- \(75\)
- \(105\)
Answer: D: \(105\)
\(2+3+7=12\) parts share \(180^\circ\): \(15^\circ\) per part. Largest \(=7\times15=105^\circ\).
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
Triangles \(ABC\) and \(DEF\) are similar, with side \(AB=6\) corresponding to side \(DE=15\). The area of triangle \(ABC\) is 20. What is the area of triangle \(DEF\)?
Show the answer and solution
Answer: 125
The scale factor is \(\tfrac{15}{6}=2.5\), so areas scale by \(2.5^2=6.25\): \(20\times6.25=125\).
Question 2
Two parallel lines are cut by a transversal. Two corresponding angles measure \((3x+10)^\circ\) and \((5x-30)^\circ\). What is the value of \(x\)?
Show the answer and solution
Answer: 20
Corresponding angles are equal: \(3x+10=5x-30\), so \(x=20\).
Question 3
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?
- \(105\)
- \(60\)
- \(50\)
- \(75\)
Show the answer and solution
Answer: D: \(75\)
\(3+4+5=12\) parts share \(180^\circ\): \(15^\circ\) per part. Largest \(=5\times15=75^\circ\).
Question 4
The measures of the angles of a triangle are in the ratio \(1:3:5\). What is the measure, in degrees, of the largest angle?
- \(60\)
- \(100\)
- \(80\)
- \(50\)
Show the answer and solution
Answer: B: \(100\)
\(1+3+5=9\) parts share \(180^\circ\): \(20^\circ\) per part. Largest \(=5\times20=100^\circ\).
Question 5
The measures of the angles of a triangle are in the ratio \(1:4:7\). What is the measure, in degrees, of the largest angle?
- \(75\)
- \(60\)
- \(105\)
- \(70\)
Show the answer and solution
Answer: C: \(105\)
\(1+4+7=12\) parts share \(180^\circ\): \(15^\circ\) per part. Largest \(=7\times15=105^\circ\).
Question 6
The measures of the angles of a triangle are in the ratio \(1:1:4\). What is the measure, in degrees, of the largest angle?
- \(120\)
- \(30\)
- \(60\)
- \(40\)
Show the answer and solution
Answer: A: \(120\)
\(1+1+4=6\) parts share \(180^\circ\): \(30^\circ\) per part. Largest \(=4\times30=120^\circ\).
Question 7
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 \(=8\times12=96^\circ\).
Question 8
Triangles \(PQR\) and \(STU\) are similar, with side \(PQ=5\) corresponding to side \(ST=10\). The area of triangle \(PQR\) is 12. What is the area of triangle \(STU\)?
Show the answer and solution
Answer: 48
The scale factor is \(\tfrac{10}{5}\), so areas scale by \(\left(\tfrac{10}{5}\right)^2\): \(12\times\tfrac{100}{25}=48\).
Question 9
Triangles \(PQR\) and \(STU\) are similar, with side \(PQ=4\) corresponding to side \(ST=6\). The area of triangle \(PQR\) is 20. What is the area of triangle \(STU\)?
Show the answer and solution
Answer: 45
The scale factor is \(\tfrac{6}{4}\), so areas scale by \(\left(\tfrac{6}{4}\right)^2\): \(20\times\tfrac{36}{16}=45\).
Question 10
Triangles \(PQR\) and \(STU\) are similar, with side \(PQ=8\) corresponding to side \(ST=12\). The area of triangle \(PQR\) is 24. What is the area of triangle \(STU\)?
Show the answer and solution
Answer: 54
The scale factor is \(\tfrac{12}{8}\), so areas scale by \(\left(\tfrac{12}{8}\right)^2\): \(24\times\tfrac{144}{64}=54\).
Question 11
Triangles \(PQR\) and \(STU\) are similar, with side \(PQ=2\) corresponding to side \(ST=7\). The area of triangle \(PQR\) is 12. What is the area of triangle \(STU\)?
Show the answer and solution
Answer: 147
The scale factor is \(\tfrac{7}{2}\), so areas scale by \(\left(\tfrac{7}{2}\right)^2\): \(12\times\tfrac{49}{4}=147\).
Question 12
Triangles \(PQR\) and \(STU\) are similar, with side \(PQ=5\) corresponding to side \(ST=10\). The area of triangle \(PQR\) is 36. What is the area of triangle \(STU\)?
Show the answer and solution
Answer: 144
The scale factor is \(\tfrac{10}{5}\), so areas scale by \(\left(\tfrac{10}{5}\right)^2\): \(36\times\tfrac{100}{25}=144\).
Keep going
- Previous topic: Area and volume
- Next topic: Right triangles and trigonometry
- All SAT Math skills · Adaptive SAT Math module simulator · Official SAT practice (Bluebook, Khan Academy)
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