Geometry: Angles, Proofs and the Sine and Cosine Laws

Angle and triangle properties, spotting the error in a proof, and solving triangles with the sine and cosine laws.

12 questions, free, marked as you go with a worked explanation on every one.

What this covers

The questions

  1. What is the sum of the interior angles of a regular heptagon (a polygon with 7 sides), in degrees?
  2. In triangle ABC, side a = 8 cm, side b = 11 cm, and the angle between them, angle C, is 40°. Find the length of side c, in centimetres, rounded to one decimal place.
  3. In triangle ABC, angle A = 35°, side a = 12 cm and angle B = 50°. Find the length of side b, in centimetres, rounded to one decimal place.
  4. Two parallel lines are cut by a transversal. One of the angles formed measures 118°. Which statement about the co-interior (same-side interior) angle paired with it is correct?
  5. In a triangle, the second angle is twice the first and the third angle is 20° more than the first. What is the measure of the smallest angle, in degrees?
  6. A student writes: 'All rectangles are parallelograms. ABCD is a parallelogram. Therefore ABCD is a rectangle.' What is wrong with this argument?
  7. A triangular garden plot has sides of 9 m, 12 m and 15 m. Using the cosine law, find the measure of the largest angle in the plot, in degrees.
  8. A surveyor stands at point C. The distance from C to A is 340 m, the distance from C to B is 260 m, and the angle ACB between those two lines of sight is 68°. How far apart are A and B, in metres, rounded to the nearest metre?
  9. A triangle has sides 7 m and 10 m with an included angle of 50° between them. Use the cosine law to find the length of the third side, in metres, rounded to one decimal place.
  10. In a triangle, side a = 8 cm is opposite a 40° angle, and side b = 11 cm is opposite angle B. Use the sine law to find the measure of angle B, in degrees, rounded to one decimal place.
  11. In a triangle, angle A = 35 degrees, side a = 10 cm and angle B = 70 degrees. Use the sine law to find side b, to the nearest tenth of a centimetre.
  12. A triangle has sides a = 7 cm, b = 9 cm and c = 12 cm. Use the cosine law to find angle A, to the nearest tenth of a degree.

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