Grade 4 Angles Practice

What this problem type covers

Sixteen problem types across five levels: angles on a straight line, angles in a triangle, three angles given by their differences, the angle between clock hands, the hours at which the hands make a given angle, a rectangle cut by a diagonal, a parallelogram, a pair of set squares meeting on a line, counting every obtuse angle in a divided figure, reading a clock forward by a turn of the minute hand, splitting a line into equal angles, two isosceles triangles standing on one line, the angle sum of a polygon with right angles, a point inside a triangle joined to two vertices, the exterior angles of a polygon, the meeting point of two angle bisectors, and a quadrilateral whose two unknown angles differ by a stated amount.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   A ray is drawn on a straight line. Find the angle ㉮.
   
   • ①128°
   • ②138°
   • ③52°
   • ④38°
   • ⑤308°
   Show solution

   ①128°

   1. Angles on a straight line add to 180°.
   2. So subtract: 180° − 52°.
   3. The size of an angle does not depend on how long the rays are drawn.
   4. ㉮ is 128°.
2. 2 · ★2
   In a triangle with angles A, B and C, B is 38° larger than A and C is 49° larger than A. How large is the biggest of the three angles?
   • ①31°
   • ②100°
   • ③80°
   • ④90°
   • ⑤69°
   Show solution

   ③80°

   1. Write all three angles in terms of the smallest one, A.
   2. A + (A + 38°) + (A + 49°) = 180°.
   3. Three lots of A come to 180° − 38° − 49°, so A is 31°.
   4. The biggest angle adds the larger difference: 80°.
3. 3 · ★3
   Two set squares are placed at one point on a straight line as shown. Find the angle ㉮.
   
   • ①135°
   • ②30°
   • ③90°
   • ④60°
   • ⑤45°
   Show solution

   ⑤45°

   1. Angles on a straight line add to 180°.
   2. Two of the three angles at the point are 45° and 90°.
   3. 180° − 45° − 90° = 45°.
   4. ㉮ is 45°.
4. 4 · ★4
   Two isosceles triangles stand on a straight line, each with two equal base angles. Find the angle ㉮.
   
   • ①112°
   • ②57°
   • ③56°
   • ④66°
   • ⑤68°
   Show solution

   ⑤68°

   1. The left apex is 66°, so each base angle is (180° − 66°) ÷ 2 = 57°.
   2. Along the line, 57° + 67° + (right base angle) = 180°.
   3. So the right base angle is 56°.
   4. That triangle is isosceles too, so ㉮ = 180° − 56° − 56° = 68°.
5. 5 · ★5
   From a point inside the triangle, lines are drawn to two of its vertices. Find the angle ㉮.
   
   • ①120°
   • ②75°
   • ③70°
   • ④110°
   • ⑤78°
   Show solution

   ④110°

   1. Split the figure into triangles and use the 180° sum in each.
   2. Apart from the angle at the inner point, the remaining angles are 43°, 35° and 32°.
   3. The angles counted twice cancel, leaving just their sum.
   4. ㉮ = 43° + 35° + 32° = 110°.

Where people usually go wrong

• Subtracting from 90 instead of 180. Angles on a straight line add to 180°.
• Thinking a longer ray means a bigger angle. The size depends only on how far the rays are opened.
• Counting only the smallest angles in a divided figure. Angles spanning several parts count too.
• Using 360° for a triangle. The three angles add to 180°.
• Answering with the smallest angle when the largest was asked for. Write all three in terms of the smallest first.
• Reading a turn of the minute hand as minutes. The minute hand turns 6° a minute, so 150° is 25 minutes.
• Halving 180° instead of what is left. Take the known angle away first, then split the remainder.
• Forgetting to remove the right angles from a polygon total before answering.
• Adding only two of the three angles for a point inside a triangle. The marked angle is the sum of all three.
• Answering 360° for the exterior angles when some of them were already given. Take the given ones away.
• Forgetting to double the halves when two bisectors meet. The bisected angles are twice what the small triangle shows.
• Forgetting the hour hand keeps moving between the hours. At half past, it sits halfway to the next number.