Educerie · SAT · Math
Geometry and Trigonometry · GT.2 Lines, angles and triangles
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Use angle sums | On a line, round a point, in a triangle |
| Use parallel-line rules | Corresponding, alternate, co-interior |
| Use the exterior-angle rule | Equal to the two opposite interior angles |
| Work with similar triangles | Equal angles, proportional sides |
| Recognise isosceles and equilateral triangles | Equal sides mean equal base angles |
1The idea in one paragraph
Every question in this unit is a short chain of two or three angle facts. There is no trick and no formula sheet needed — just the handful of rules below, applied in order until the marked angle appears. The most useful habit is writing each angle you find onto the figure as you find it, because the chain is usually shorter than it looks once two angles are filled in.
Diagrams on this test are not drawn to scale unless it says so. Never measure; never judge by eye which angle looks bigger.
2The angle facts
| Fact | Value |
|---|---|
| Angles on a straight line | 180° |
| Angles round a point | 360° |
| Angles in a triangle | 180° |
| Angles in a quadrilateral | 360° |
| Vertically opposite angles | Equal |
| Exterior angle of a triangle | Sum of the two opposite interior angles |
The exterior-angle rule is the one that saves the most time: instead of finding the third interior angle and subtracting from 180, take the two you were given and add them.
3Parallel lines
When a line crosses two parallel lines:
- Corresponding angles (same position at each crossing) are equal
- Alternate angles (opposite sides of the crossing line, between the parallels) are equal
- Co-interior angles (same side, between the parallels) sum to 180°
In practice: at each crossing there are only two distinct angle sizes, and they add to 180°. Find one and you have them all.
4Triangles by their sides
| Triangle | Sides | Angles |
|---|---|---|
| Equilateral | All three equal | All 60° |
| Isosceles | Two equal | The two opposite them are equal |
| Scalene | All different | All different |
The isosceles rule runs both ways: equal sides give equal angles, and equal angles give equal sides. A question marking two sides with the same tick is telling you two angles are equal.
5Similar triangles
Two triangles with the same angles are similar: their sides are in proportion. This appears most often as a small triangle inside a large one sharing an angle, or as two triangles formed by parallel lines.
Set up the proportion with matching sides in matching positions:
Then cross-multiply. The most common error is pairing a side of one triangle with a non-matching side of the other, which gives a plausible number and a wrong one.
6The triangle inequality
Any two sides of a triangle must sum to more than the third. A question giving two sides and asking which third length is possible is asking exactly this: with sides 7 and 10, the third must be greater than 3 and less than 17.
Where points are lost
- Assuming the diagram is to scale.
- Mixing up alternate and co-interior angles, and so using equality where 180° was needed.
- Forgetting the third angle in a triangle when the exterior-angle rule would have skipped it.
- Pairing the wrong sides in a similar-triangle proportion.
- Missing an isosceles triangle because only the tick marks show it.
- Stopping at an intermediate angle rather than the one asked for.
Work it right
- Mark every angle you are given onto the figure.
- Look for parallel lines: they hand you equal or supplementary angles immediately.
- Look for isosceles marks.
- Chain the facts, writing each new angle down.
- Check you have answered for the angle the question named.
Try it
Q1. In a triangle, two angles measure 47° and 68°. What is the measure of the third angle, in degrees? (Type your answer.)
Q2. Two parallel lines are crossed by a third line. One of the eight angles formed measures 115°. Which is not a possible measure of another of the eight angles?
A) 65° B) 115° C) 125° D) None of the other angles differ from these
Q3. An exterior angle of a triangle measures 130°, and one of the two opposite interior angles measures 55°. What is the other opposite interior angle, in degrees? (Type your answer.)
Q4. Triangle ABC is similar to triangle DEF. AB = 6, BC = 9 and DE = 8. What is the length of EF?
A) 10.5 B) 12 C) 13.5 D) 15
Q5. Two sides of a triangle measure 9 and 4. Which could be the length of the third side?
A) 4 B) 5 C) 6 D) 13
In one breath
Write every angle you find onto the figure and chain the rules: 180° on a line and in a triangle, 360° round a point, and at a crossing of parallel lines only two angle sizes exist and they add to 180°. The exterior-angle rule saves a step whenever it applies, tick marks are how the test tells you a triangle is isosceles, and a similar-triangle proportion only works when matching sides sit in matching positions.
Answers
Q1. 65. angles in a triangle sum to 180° 180 − 47 − 68 = 65.
Q2. C — 125°. only two sizes exist, and they are supplementary The eight angles are all either 115° or 65°. A and B are those two; D contradicts the rule.
Q3. 75. the exterior angle equals the sum of the two opposite interiors 130 − 55 = 75. Finding the adjacent interior angle first (50°) and subtracting gives the same answer by a longer route.
Q4. B — 12. matching sides in matching positions
C comes from pairing the sides the other way round.
Q5. C — 6. the third side lies strictly between 5 and 13
A and B are too short; D equals the sum exactly, which gives a flat line rather than a triangle.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Geometry and Trigonometry, skill/knowledge testing point "Lines, angles, and triangles"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
Check your understanding
The main ideas of this note. Tick each one you could do now, in an exam, without looking back up. Anything you cannot tick yet is the part to read again.