SAT Parallel Lines and Angles: Check on Desmos

When a line crosses two parallel lines, corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to 180 degrees. On the SAT you pick the right relationship, write one equation and solve for x. Then you can graph both sides on Desmos, click where they cross, and check the angle measures.

You can run each check below in the College Board version of Desmos on our Desmos on the SAT practice calculator page. This guide gives you the angle rules, one worked example, one guided example, one problem to try on your own and two harder SAT-style problems, each with its answer.

The rule: eight angles, two sizes

A transversal is a line that crosses two other lines. When it crosses two parallel lines, it makes eight angles. Unless the transversal is perpendicular, those eight angles come in only two sizes: four acute angles with the same measure and four obtuse angles with the same measure. Any one of the acute angles plus any one of the obtuse angles is 180°.

To name the pairs, number the angles. At the upper crossing, call the top-left angle 1, the top-right angle 2, the bottom-left angle 3 and the bottom-right angle 4. At the lower crossing, use 5, 6, 7 and 8 for the same four spots. In every example on this page the transversal slants up to the right, so angles 2, 3, 6 and 7 are acute and angles 1, 4, 5 and 8 are obtuse.

That is seven named pairs: four that are equal and three that add to 180°. Vertical angles and linear pairs work wherever two lines cross. The other five need the two lines to be parallel, so a question has to tell you that they are. If a question doesn't say the lines are parallel, you can't use those five.

A quick test covers most questions. Two acute angles are equal, and so are two obtuse angles. One acute angle and one obtuse angle add to 180°. Decide which is which from the positions in the table, not from how the drawing looks.

Worked example (I do): an equal pair

SAT-style question. Two parallel lines are cut by a transversal. Angle 3 measures (3x + 15)° and angle 6 measures (5x - 25)°. What is the value of x, and what is the measure of angle 3?

Step 1. Name the pair. Angles 3 and 6 sit between the parallel lines, on opposite sides of the transversal. They are alternate interior angles, so they are equal. Both are acute in our figure, which fits.

Step 2. Write the equation. 3x + 15 = 5x - 25.

Step 3. Graph each side as its own line. Type:

1  y = 3x + 15
2  y = 5x - 25

I graph both sides because the two lines cross at the one value of x that makes the sides equal. Desmos's help page says that when you click a line, gray dots appear at intercepts and intersection points, and clicking a gray dot shows its coordinates. Click either line and select the gray dot where the lines cross. It reads (20, 75).

Step 4. Read both numbers. The x-coordinate, 20, is x. The y-coordinate, 75, is the value of each side, so it is the angle measure. Angle 3 is 75°, and so is angle 6.

By-hand check. Subtract 3x from both sides and add 25: 40 = 2x, so x = 20. Then 3(20) + 15 = 75 and 5(20) - 25 = 75. The two angles match, and 75° is acute, as angle 3 should be.

Guided example (we do): a pair that adds to 180°

SAT-style question. Two parallel lines are cut by a transversal. Angle 4 measures (5x + 12)° and angle 6 measures (3x + 8)°. What is the measure of angle 4?

Step 1. Angles 4 and 6 are both between the parallel lines and on the same side of the transversal. They are same-side interior angles, so they add to 180°. Angle 4 is obtuse and angle 6 is acute, which fits a pair that adds to 180°.

Step 2. The equation is (5x + 12) + (3x + 8) = 180. To make Desmos show angle 4 directly, move angle 6 to the other side: 5x + 12 = 180 - (3x + 8).

Step 3. Type the two sides:

1  y = 5x + 12
2  y = 180 - (3x + 8)

Keep the parentheses on line 2. Without them Desmos reads 180 - 3x + 8, which is a different line.

Step 4 (your step). Click a line, select the gray dot where the lines cross, and read off x and the measure of angle 4. Then find angle 6.

Check your answer: The lines cross at (20, 112). So x = 20 and angle 4 is 112°. Angle 6 is 3(20) + 8 = 68°, and 112 + 68 = 180.

Your turn (you do)

SAT-style question. Two parallel lines are cut by a transversal. Angle 1 measures (7x - 18)° and angle 7 measures (3x + 28)°. What is the measure of angle 1?

Find the pair in the table first. Then decide whether the two expressions are equal or add to 180°, graph it and check by hand. Try it before you read the next paragraph.

Answer: 101°. Angles 1 and 7 are same-side exterior angles, so (7x - 18) + (3x + 28) = 180. That gives 10x + 10 = 180 and x = 17. Angle 1 is 7(17) - 18 = 101°, and angle 7 is 3(17) + 28 = 79°. If you set the two expressions equal instead, you get x = 11.5 and 62.5° for both angles. That can't be right, because angle 1 is obtuse.

Two harder SAT-style problems

Stretch 1: two unknowns

SAT-style question. Two parallel lines are cut by a transversal. Angle 1 measures (2x + y)°, angle 5 measures (x + 4y)° and angle 2 measures (x + 2y)°. What is the value of x?

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Angles 1 and 5 are corresponding angles, so 2x + y = x + 4y, which simplifies to x = 3y. Angles 1 and 2 form a linear pair, so (2x + y) + (x + 2y) = 180, which simplifies to x + y = 60. Now you have a system of two equations. Solve each one for y and graph both:

1  y = x/3
2  y = 60 - x

The lines cross at (45, 15), so x = 45 and y = 15. This time the graph's x and y are the two unknowns from the question, not x and an angle measure. By hand, put x = 3y into x + y = 60 to get 4y = 60, so y = 15 and x = 45. Check the angles: angle 1 is 2(45) + 15 = 105°, angle 5 is 45 + 4(15) = 105° and angle 2 is 45 + 2(15) = 75°. The corresponding angles match, and 105 + 75 = 180. For more problems like this, see how to solve systems of equations on Desmos.

Stretch 2: a parallel line through a triangle

SAT-style question. Triangle ABC has side AC along the bottom. Line ℓ passes through vertex B and is parallel to side AC. The angle between line ℓ and side BA, on the same side of B as vertex A, measures 55°. Angle ABC measures 80°. What is the measure of angle BCA?

Side BA is a transversal that crosses the parallel lines ℓ and AC. The 55° angle at B and angle BAC are alternate interior angles, so angle BAC is 55°. The angles of a triangle add to 180°, so angle BCA = 180 - 55 - 80 = 45°. To check, look at the three angles that sit along line ℓ at B: 55°, 80° and 45°. Together they make a straight line, 180°.

You don't need Desmos here. The work is naming the pair, and the arithmetic takes one line.

When algebra is faster

College Board allows a calculator on the entire Math section, and Bluebook has a Desmos calculator built in. College Board's calculator policy also notes that some Math questions are better done without a calculator, even though you're allowed to use one. We put most one-variable angle equations in that group: one subtraction and one division give you x. Graph it when the expressions are messy, when you want to confirm an answer before you move on, or when the question has two unknowns, as in Stretch 1.

Mistakes to avoid

• Setting a 180° pair equal. If your two angles come out the same size but one of them should be obtuse, you used the wrong rule. Decide whether each angle is acute or obtuse from its position before you type anything.
• Answering x instead of the angle. The guided example asks for angle 4, which is 112°, not x = 20. When you graph y = an angle expression, the y-coordinate of the crossing is the angle.
• Dropping parentheses. Type y = 180 - (3x + 8), not y = 180 - 3x + 8. The second version crosses y = 5x + 12 at (22, 122), which gives a wrong angle.

Practice next

Work through the free Angles lesson notes, then find more geometry questions in the SAT question bank. For other geometry topics on Desmos, read circle tangent slope with a Desmos check and SAT volume problems on Desmos.

Checked on September 26, 2026 against College Board's SAT Suite calculator policy, College Board's March 12, 2024 digital SAT launch release, the College Board version of the Desmos graphing calculator and the Desmos Help Center help article on finding points on a graph. Every answer on this page was recomputed before publishing. The Test Advantage is independent of College Board and Desmos.