Lines, angles and triangles

Digital SAT · Math · Geometry and Trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

These questions give a figure, or describe one in words, and ask for a missing angle or side. If there is no figure, draw one and mark everything you are told.

Almost every angle question is solved with a few facts: angles on a straight line, angles made by parallel lines, and the angle sum of a triangle. Write an equation from one of these facts and solve it.

Similar triangles have the same three angles, so one is an enlarged copy of the other. Their matching sides are in the same ratio. A line drawn inside a triangle parallel to one side always makes a smaller triangle similar to the whole triangle.

Rules to remember

Common mistake

In a triangle with a line parallel to the base, comparing the top part of a side with the bottom part. The ratio of similar triangles uses whole sides: the small triangle's side over the full side of the large triangle.

Worked example

In triangle PQR, the measure of angle P is (2x + 10)°, the measure of angle Q is 3x°, and the measure of angle R is (x + 20)°. What is the measure of the largest angle?

  1. Angle sum: (2x + 10) + 3x + (x + 20) = 180, so 6x + 30 = 180.
  2. 6x = 150, so x = 25.
  3. Angles: P = 2(25) + 10 = 60°, Q = 3(25) = 75°, R = 25 + 20 = 45°. Check: 60 + 75 + 45 = 180.

Answer: 75° (angle Q)

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