SAT Math · Geometry and Trigonometry · Skill 2 of 4

Lines, angles and triangles

Use the angle rules for straight lines, parallel lines, triangles and polygons, then find missing sides with similar triangles and tell when two triangles must match.

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Hi, I’m Io. Almost every angle question comes down to two numbers: 180 and 360. Let me show you where they hide.
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Parallel lines and a transversal

Explore · 5 min

Lines ℓ and m are parallel. The slanted line crossing both is a transversal, and it makes eight angles. Tap any two angles to see how they are related, then tilt the transversal with the slider.

The basic angle rules

Learn · 4 min

Straight lineAngles on a line add to 180°
Vertical anglesOpposite angles at a crossing are equal
Around a pointAngles all the way round add to 360°

Angles with parallel lines

Practise · 5 min

When a transversal crosses two parallel lines, only two different sizes of angle appear. Every angle is either the small one or the big one, and small + big = 180°.

  • Equal pairs: vertical, corresponding (same position at each crossing), alternate interior (between the lines, opposite sides), alternate exterior
  • Pairs that add to 180°: same-side interior (between the lines, same side), and any two angles next to each other on a line

Angles in triangles

Learn · 6 min

Angle suma + b + c = 180°
Exterior angled = a + c, the two angles not next to it
IsoscelesTwo equal sides, so the two base angles are equal
  • An equilateral triangle has three 60° angles
  • The longest side is opposite the largest angle
  • Any side must be shorter than the other two added together, or the triangle cannot close

Angles in polygons

Learn · 4 min

From one corner, a polygon with n sides splits into n − 2 triangles, and each triangle gives 180°.

Interior angle sum = (n − 2) × 180°
  • In a regular polygon, all angles are equal: divide the sum by n
  • The exterior angles of any polygon add to 360°, one at each corner

Similar triangles

Practise · 6 min

Triangles are similar when they have the same angles. Then they are the same shape at different sizes, and every pair of matching sides has the same ratio. Two pairs of equal angles is enough (the third must match too).

Worked example: in triangle ABC, DE is parallel to BC. AD = 3, DB = 6 and DE = 4. Find BC.
  1. DE is parallel to BC, so triangle ADE is similar to triangle ABC (they share angle A, and the parallel lines make the other angles match)
  2. Match sides: AD goes with AB. AB = 3 + 6 = 9, so the scale factor is 9 ÷ 3 = 3
  3. DE goes with BC: BC = 4 × 3 = 12

Use the whole sideAB is AD + DB, not DB alone. Comparing DB with AD is the most common slip here.

Congruent, similar, or not enough?

Sort · 4 min

Congruent triangles are identical: same angles and same sides. These clues prove it: SSS (three sides), SAS (two sides and the angle between them), ASA or AAS (two angles and a side). Similar triangles only need AA, or sides in the same ratio. Sort each clue.

Angles and triangles in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
LinesA straight line is 180°, a full turn is 360°.
Parallel linesSmall angles equal, big angles equal, small + big = 180°.
TrianglesAngles add to 180°. Exterior angle = the two far interior angles.
PolygonsInterior sum = (n − 2) × 180°. Exterior angles add to 360°.
SimilarSame angles, sides in one ratio. Match sides first.
CongruentSSS, SAS, ASA or AAS. SSA and AAA are not enough.
Challenge