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Angles
Angle calculation involves various operations such as addition, subtraction, multiplication, and division of angles. These calculations are often performed using trigonometric functions.
Questions: 10Estimated time: 13 minutes
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Question 1out of 10
Four angles meet around a point. Three are 60°, 70° and 90°. What is the fourth angle x?
A full turn around a point is 360°. Add the three known angles and subtract their sum from 360°.
Question 2out of 10
Triangle ABC is isosceles. Find angle ABC, knowing that angle CAB is 38°.
The base angles of an isosceles triangle are equal, and all angles of a triangle add up to 180°.
- Subtract the given vertex angle from 180°.
- Divide the remaining angle sum by 2 because the two base angles are equal.
Formula: required angle = (180° − vertex angle) ÷ 2.
Question 3out of 10
Find angle ABD, knowing that angle ADC is 94°, and angle DAB is 13°.
In the diagram, points C, D and B lie on one straight line, so the adjacent angles at D add up to 180°. The interior angles of a triangle also add up to 180°.
- Use the straight angle to find the angle of triangle ADB at D: ∠ADB = 180° − ∠ADC.
- Add the two known angles in triangle ADB.
- Subtract their sum from 180° to obtain the angle at B.
A useful check: the three angles of triangle ADB must total exactly 180°.
Question 4out of 10
Three interior angles of a quadrilateral are 90°, 90° and 90°. Find the fourth angle x. The drawing is not to scale.
The interior angles of a quadrilateral total 360°. Subtract the sum of the three known angles.
Question 5out of 10
Find the angle AOB measuring clockwise.
The circle is divided into 8 equal parts, so each interval is 360° ÷ 8 = 45°. Count clockwise intervals from OA to OB and multiply by 45°. The requested angle may exceed 180°.
Question 6out of 10
Triangle ABC is isosceles. Find angle CAB, knowing that angle ABC is 34°.
The equally marked sides AB and AC are equal, so the base angles at B and C are equal. Triangle angles total 180°. The vertex angle CAB = 180° − 2 × angle ABC.
Question 7out of 10
The angles A, B and C of a triangle are in the ratio 1 : 3 : 5. Find angle A. The drawing is not to scale.
Add the ratio terms. Divide the triangle total of 180° by that sum, then multiply by the term corresponding to the requested angle.
Question 8out of 10
In the diagram, a square is depicted. Find angle ACD.
Square corners are 90°. Its diagonals bisect the corner angles and meet at right angles. The drawing therefore contains 45°, 90°, 135° and 180° angles. The middle letter of an angle name identifies its vertex; locate its two rays and add or subtract known angles.
Question 9out of 10
In the diagram, a rectangle is depicted. Find angle AOD, knowing that angle ADO is 57°.
In triangle ADO, the angle at A is a right angle of 90°. Triangle angles total 180°, so angle AOD = 180° − 90° − angle ADO.
Question 10out of 10
In the diagram, a rectangle is depicted. Find angle DOB, knowing that angle ADO is 58°.
Every corner of a rectangle is 90°. First use the triangle formed by a side of the rectangle, its top side and the slanted segment in the diagram.
- Subtract the given angle from 90° to find the other acute angle at O.
- Points A, O and B lie on one straight line, so the angles formed with rays OA and OB are supplementary.
- Find the requested angle by subtracting the angle found from 180°.
Do not confuse the rectangle’s 90° corner with the full angle at O: that angle depends on which side of the straight line the rays bound.
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Sample questions
- Find the angle AOB measuring clockwise.
- Triangle ABC is isosceles. Find angle ABC, knowing that angle CAB is ___°.
- Triangle ABC is isosceles. Find angle CAB, knowing that angle ABC is ___°.
- Find angle ABD, knowing that angle ADC is ___°, and angle DAB is ___°.
- In the diagram, a rectangle is depicted. Find angle AOD, knowing that angle ADO is ___°.
Angle calculations
Angles on a straight line total 180°, angles around a point total 360°, and parts of a right angle total 90°. Identify the relationship shown before calculating.
Triangles and intersecting lines
The angles of a triangle total 180°, and vertically opposite angles formed by intersecting lines are equal. A diagram may not be to scale, so use labels and stated values.
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