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Lines, angles, and triangles: SAT Math notes
Use angle facts about crossing lines, parallel lines and triangles to find missing angles, and use similar or congruent triangles to find missing sides.
Study it with worked examples and 8 practice questionsPractise Lines, angles, and triangles questions
What this topic tests
Most of these questions come with a figure: lines crossing, two parallel lines cut by a third line, or one or two triangles. The work is a chain. One fact gives you an angle, that angle unlocks the next one, and you keep going until you reach the angle you were asked for.
A handful of facts do almost all the work. Angles on a straight line add to 180°. Vertical angles are equal. Parallel lines create equal angles. A triangle's angles add to 180°, which is on the reference sheet.
The other half of the topic is similar and congruent triangles. Similar triangles have the same shape, so their angles are equal and their sides are in proportion. Congruent triangles are exact copies. You'll either use that to find a length, or decide which facts are enough to prove two triangles match.
Figures are often marked not drawn to scale. Trust the numbers and the facts you're given, not how the picture looks.
Key ideas
- Angles that make a straight line add to 180°. Angles all the way around a point add to 360°.
- Vertical angles, the ones opposite each other where two lines cross, are equal.
- When a line crosses two parallel lines, every angle it makes is one of just two sizes, and those two sizes add to 180°.
- A triangle's angles add to 180°. An exterior angle equals the sum of the two inside angles that aren't next to it.
- In an isosceles triangle, the angles opposite the equal sides are equal.
- Similar triangles have equal matching angles and matching sides in the same ratio. Scaling a shape changes its sides but never its angles.
- Two pairs of equal angles (AA) prove that triangles are similar. Two sides in the same ratio are not enough unless the angle between them matches too.
The 4 kinds of Lines, angles, and triangles question
Sorted from every Lines, angles, and triangles question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Angles at crossing and parallel lines
How to spot it. The figure shows lines crossing, often two parallel lines cut by a third line called a transversal. You're asked for an angle, or which extra fact would prove that two lines are parallel.
How to answer it:
- Find where the known angle and the unknown angle sit. Are they at the same crossing point or at different ones?
- At one crossing point: vertical angles are equal, and side-by-side angles on a line add to 180°.
- Across parallel lines: angles in the same position at each crossing (corresponding angles) or in a Z shape (alternate angles) are equal. Two angles between the parallel lines on the same side of the transversal add to 180°.
- Quick check: if both angles are acute, or both obtuse, they're equal. If one is acute and the other obtuse, they add to 180°.
- For a question about which fact proves the lines are parallel, pick the choice that makes one of these relationships true.
The trap. Saying two angles are equal when they actually add to 180°, or the other way round. Before you write an equation, look at the two angles: one small and one big means they add to 180°. Also, a fact that is always true, like a pair of vertical angles being equal, can't prove that lines are parallel.
2. Angle chasing in triangles
How to spot it. There's at least one triangle, sometimes several sharing sides or crossed by other lines. You know some of the angles, and maybe which sides are equal, and you need another angle.
How to answer it:
- Mark every angle you already know on the figure, including the ones you get for free from straight lines and vertical angles.
- Look for a triangle with two known angles. Its third angle is 180 minus the other two.
- If two sides are equal, the angles opposite them are equal. That often hands you a second angle.
- For an angle outside the triangle, on an extended side, use the exterior angle rule: it equals the sum of the two far inside angles.
- Keep moving from triangle to triangle until you reach the angle you need.
The trap. Stopping at the wrong angle. In a figure with several steps you'll find a few angles on the way, and each of them is often a wrong choice. Before you answer, check that your angle is the one marked with the letter.
3. Similar triangles: finding a length
How to spot it. Two triangles have the same shape but different sizes. Common setups are shadows cast at the same time of day, a line drawn parallel to one side of a triangle, and two triangles meeting at a point between parallel lines. You need a missing side, or sometimes an area.
How to answer it:
- Show that the triangles are similar by finding two equal angles: a shared angle, vertical angles, right angles, or angles made by parallel lines.
- Match the corners. Sides opposite equal angles correspond.
- Write a proportion with matching sides, keeping the same order on both sides: small/big=small/big.
- Cross multiply and solve. If the question asks about areas, square the side ratio.
- If you need only part of a side, subtract the piece you found from the whole side.
The trap. Matching the wrong sides. Sides that look alike in the figure don't always correspond, so match them by the angles they're opposite. And for areas, remember the area ratio is the side ratio squared.
4. Proving similarity or congruence
How to spot it. The question asks which extra fact is enough to prove two triangles are similar or congruent. Or it tells you two shapes are similar, congruent or dilated, and asks for a matching angle or side.
How to answer it:
- List what the two shapes already have in common.
- Remember what counts as proof. Similar: two pairs of equal angles (AA), all three pairs of sides in the same ratio (SSS), or two pairs of sides in the same ratio with equal angles between them (SAS). Congruent: the same ideas, with equal sides instead of proportional ones.
- Test each choice. When a triangle has two known angles, use the angle sum to find the third, then see whether two angles really match.
- If the shapes are already known to match, pair up the corners in the order they're named. Matching angles are equal, however much the shape was scaled.
The trap. Thinking a scale factor changes angles. A triangle dilated by 3 or by 26 keeps exactly the same angles. Another trap: two sides in the same ratio plus an angle that is not between them do not prove similarity.
Practise Lines, angles, and triangles
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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