On the test
Geometry and Trigonometry is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I look at a question about lines, angles or triangles, I think about a map of roads. Imagine two straight roads that never meet, running side by side. A third road crosses both. Every crossing makes the same shapes. If you know the turn at the first crossing, you already know the turn at the second one, without measuring. That is the whole idea of angle rules: nature repeats itself, so one known number unlocks others. And a triangle is like a path with three turns. However you walk it, the turns always add up to the same total when you come back. The SAT asks the same kind of questions. It gives you a few angles or lengths, and you use a small number of fixed rules to find the missing one.
There are only a few angle and triangle rules, and the SAT uses the same ones again and again. Once you know them, these questions become a quick puzzle you can enjoy.
What is inside this part
Using rules about angles on a line, around a point, and where lines cross.
Supplementary and complementary angles · Vertical angles · Angles on a straight line
Finding angles when a line crosses two parallel lines.
Alternate interior angles · Corresponding angles · Same-side interior angles · Solving for x
Using the angle sum, exterior angle, isosceles and side-length rules of a triangle.
Angle sum of 180° · Exterior angles · Isosceles and equilateral triangles · Triangle inequality
Using equal angles and matching side ratios to find missing lengths.
Scale factor · Line parallel to one side · Perimeter and area ratios
How to know a question is from this part
A figure with lines, angles or triangles is described, with some angles or sides given and one missing.
- Words like parallel, transversal, supplementary, complementary, vertical angles
- 'Triangle ABC is similar to triangle DEF'
- 'Line DE is parallel to side BC'
- Angles written with x, like (3x + 10)°
- 'Which of the following could be the length of the third side?'
The lesson
1. Lines and angles: the basic words
An angle is the opening between two lines that meet at a point. We measure it in degrees (°). A right angle is exactly 90°. A straight line is 180°. A full turn around a point is 360°.
Two angles that add to 90° are called complementary. Two angles that add to 180° are called supplementary. Angles that sit side by side on a straight line add to 180°.
When two straight lines cross, they make four angles. Angles that are opposite each other (vertical angles) are equal.
2. Parallel lines and a transversal
Parallel lines are lines that never meet, like the two rails of a railway. A transversal is a line that crosses both of them. This makes eight angles, but there are only two sizes: one small size and one big size. The small angles are all equal to each other, the big ones are all equal to each other, and any small plus any big makes 180°.
The names you may see: corresponding angles (same position at each crossing) are equal. Alternate interior angles (inside the parallels, on opposite sides) are equal. Same-side interior angles add to 180°.
Trick: if you can see which angles look small and which look big, you only need to decide equal or add to 180°. Equal if both are small or both are big. Add to 180° if one is small and one is big.
Two parallel lines are cut by a transversal. One angle is (3x + 10)° and the alternate interior angle is (5x − 30)°. What is the size of each angle?
Show the steps
- Alternate interior angles are equal: 3x + 10 = 5x − 30.
- Solve: 40 = 2x, so x = 20.
- Put back in: 3(20) + 10 = 70.
70°
3. Triangles: the angle rules
The three angles inside any triangle add to 180°. If you know two, subtract them from 180° to get the third.
An exterior angle is the angle outside the triangle, made by extending one side. It equals the sum of the two inside angles that are not next to it. It also equals 180° minus its neighbouring inside angle.
In an isosceles triangle, two sides are equal and the two angles opposite those sides (the base angles) are equal. In an equilateral triangle, all sides are equal and each angle is 60°.
In triangle ABC, the exterior angle at C is 130° and angle A is 50°. What is angle B?
Show the steps
- The exterior angle equals the sum of the two far angles: 130 = 50 + B.
- B = 80.
80°
4. Side rules in a triangle
The longest side is always opposite the biggest angle, and the shortest side is opposite the smallest angle.
Triangle inequality: any side must be shorter than the other two added together, and longer than their difference. If two sides are 7 and 12, the third side must be between 12 − 7 = 5 and 12 + 7 = 19, and cannot be exactly 5 or 19.
Two sides of a triangle are 7 and 12. Which of these could be the third side: 4, 5, 12, 19?
Show the steps
- The third side must be bigger than 12 − 7 = 5.
- It must be smaller than 12 + 7 = 19.
- Only 12 is between 5 and 19.
12
5. Similar triangles
Similar triangles have the same shape but not necessarily the same size. Their matching angles are equal, and their matching sides are in the same ratio. This ratio is called the scale factor.
If triangle ABC is similar to triangle DEF, and AB matches DE, then AB/DE = BC/EF = AC/DF. Write the matches carefully from the order of the letters.
Perimeters of similar triangles have the same ratio as the sides. Areas have the ratio squared.
Triangle ABC is similar to triangle DEF. AB = 6, BC = 8, AC = 7, and DE = 15. What is the perimeter of DEF?
Show the steps
- Scale factor = DE/AB = 15/6 = 2.5.
- Perimeter of ABC = 6 + 8 + 7 = 21.
- Perimeter of DEF = 21 × 2.5 = 52.5.
52.5
6. A line parallel to a side
When a line is drawn inside a triangle parallel to one side, it cuts off a smaller triangle that is similar to the big one. This is a very common SAT picture.
Be careful: the small triangle's side matches the whole big side, not the cut-off piece. For example, if D is on AB and E is on AC with DE parallel to BC, then triangle ADE is similar to ABC, and AD/AB = DE/BC. The whole AB is AD + DB.
7. Common traps
Trap 1: using a side piece (like DB) instead of the whole side (AB) in a similarity ratio. Trap 2: assuming two lines are parallel just because they look parallel; the question must say so. Trap 3: assuming a triangle is isosceles or right because of the drawing.
Trap 4: forgetting that angles on a line add to 180°, not 90°. Trap 5: mixing up supplementary (180°) and complementary (90°). Trap 6: solving for x and then answering with x when the question asks for the angle.
8. Ratios of angles
Sometimes angles are given in a ratio, like 2:3:4. Add the parts (2 + 3 + 4 = 9), then divide the total (180° for a triangle) by 9 to get one part (20°). Multiply to get each angle: 40°, 60°, 80°.
The angles of a triangle are in the ratio 2:3:4. What is the difference between the largest and smallest angle?
Show the steps
- One part = 180 ÷ 9 = 20°.
- Largest = 4 × 20 = 80°, smallest = 2 × 20 = 40°.
- Difference = 40°.
40°
See it
- Both are small, or both are big
- Alternate interior angles
- Corresponding angles
- Vertical angles (opposite at a crossing)
- One is small and one is big
- Same-side interior angles
- Two angles side by side on a straight line
- Supplementary angles
| Fact | Rule |
|---|---|
| Inside angles | Add to 180° |
| Exterior angle | Equals the two far inside angles added |
| Isosceles | Two equal sides, two equal angles opposite them |
| Equilateral | Three equal sides, three angles of 60° |
| Third side | Between the difference and the sum of the other two |
| Similar triangles | Equal angles, sides in the same ratio |
DE is parallel to BC, so triangle ADE is similar to triangle ABC. Compare AD to the whole AB, not to DB.
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