Lines, angles, and triangles
Geometry and Trigonometry is about 15% of SAT Math (5–7 of 44 questions). Angle and triangle questions use a figure or a short description, and often ask you to set up and solve a simple equation for x.
Angles on lines
Angles on a straight line add up to 180°. Angles around a point add up to 360°. When two lines cross, the angles opposite each other are called vertical angles, and they are equal. Two neighboring angles on a line are supplementary. These three facts alone solve many SAT questions: write them as equations and solve for x.
Parallel lines and a transversal
When a transversal crosses two parallel lines, it makes eight angles but only two different sizes. In the figure, all angles marked a are equal, all angles marked b are equal, and a + b = 180°. Angles that are in the same position at the two crossings (corresponding) are equal. Angles on opposite sides inside the parallel lines (alternate interior) are equal. Angles on the same side inside the lines (same-side interior) add to 180°.
Triangle facts
The three angles of a triangle add up to 180°. An exterior angle equals the sum of the two interior angles that are not next to it. In an isosceles triangle, the angles opposite the two equal sides are equal. An equilateral triangle has three equal sides and three 60° angles. The longest side is opposite the largest angle.
| Fact | Rule |
|---|---|
| Angle sum | A + B + C = 180° |
| Exterior angle | exterior = sum of the two far interior angles |
| Isosceles | equal sides ↔ equal opposite angles |
| Equilateral | all sides equal, all angles 60° |
Similar triangles
Two triangles are similar if their angles are equal (two equal pairs are enough, because the third pair then follows). Their sides are in proportion: every side of one triangle is multiplied by the same scale factor to give the matching side of the other. A line drawn inside a triangle parallel to one side cuts off a smaller triangle that is similar to the big one. Always match corresponding sides: the shortest with the shortest, and so on.
Congruent triangles
Congruent triangles are exact copies: same angles and same sides. Three facts are enough to prove it: SSS (three sides), SAS (two sides and the angle between them), ASA or AAS (two angles and a side). Three equal angles (AAA) are not enough: that gives only similar triangles. Two sides and an angle that is not between them (SSA) is also not enough.
| Given | Congruent? |
|---|---|
| SSS | Yes |
| SAS (angle between the sides) | Yes |
| ASA or AAS | Yes |
| AAA | No: only similar |
| SSA (angle not between) | No |
- The marked angles are on the same side of the transversal, between the parallel lines: same-side interior angles.
- Same-side interior angles add up to 180°.
- (4x + 20) + (2x + 10) = 180, so 6x + 30 = 180 and 6x = 150.
- x = 25. Check: 4(25) + 20 = 120 and 2(25) + 10 = 60, and 120 + 60 = 180 ✓
- The exterior angle equals the sum of the two interior angles far from it: ∠ACD = ∠A + ∠B.
- 125 = 50 + ∠B, so ∠B = 75.
- Check with the angle sum: ∠C = 180 − 125 = 55, and 50 + 75 + 55 = 180 ✓
- There are two cases, because the 50° angle can be the odd angle or one of the two equal angles.
- Case 1: 50° is one of the two equal angles. The angles are 50°, 50° and 80°.
- Case 2: 50° is the odd angle. The other two are equal: (180 − 50) ÷ 2 = 65°, so the angles are 50°, 65°, 65°.
- The largest possible other angle is 80°.
- Because DE ∥ BC, triangle ADE is similar to triangle ABC.
- AB = AD + DB = 6 + 3 = 9.
- The proportion is AB / AD = BC / DE, so 9 / 6 = BC / 8.
- BC = 8 × 9 / 6 = 12.
- Setting same-side angles equalLook at the picture: if one angle is obtuse and the other is acute, they are supplementary (sum 180°), not equal.
- Forgetting that the figure is not always to scaleThe SAT says “not drawn to scale” on some figures. Use the facts that are given, not the way the angles look.
- Using the exterior angle as an interior oneThe exterior angle equals the sum of the two far interior angles; the interior angle next to it is 180° minus the exterior angle.
- Mismatching the sides of similar trianglesWrite the triangles with matching letters (ADE ∼ ABC) and build the proportion from that order.
- Using only part of a side in a proportionAD and DB are parts of AB. The similar triangles ADE and ABC use AD and the whole AB.
- Believing that equal angles prove congruenceAAA gives similar triangles of any size. Congruence needs at least one pair of equal sides.
Desmos cannot draw geometric figures with exact angles, but it solves the equations that come from them. Type a one-variable equation such as 4x + 20 + 2x + 10 = 180 and Desmos draws the vertical line at the solution. For similar triangles, type a proportion like 9/6 = x/8 and read the value of x.
- Type 4x + 20 + 2x + 10 = 180 and read the x-coordinate of the vertical line
- Type 9/6 = x/8 for a proportion between similar triangles
- Check the answer by typing the angle expressions with x replaced by your value
Solving by hand is faster for simple angle equations. Use Desmos when the numbers are not nice or to double-check a result.
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