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Sum of the Measures of the Exterior Angles Quiz - CBSE Class 8 Mathematics

Free practice quiz: Sum of the Measures of the Exterior Angles (CBSE Class 8 Mathematics). 17 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 17 multiple-choice questions on Sum of the Measures of the Exterior Angles, and 17 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Exterior Angle and Linear Pair

  • Extend one side of a polygon at a vertex. The angle between this extension and the next side is the exterior angle.
  • At each vertex, the interior angle and its exterior angle lie on a straight line, so they add up to 180°.
Linear pair at a vertex
\[\text{interior angle} + \text{exterior angle} = 180^\circ\]
Exterior from interior
\[e = 180^\circ - i\]

Common mistake: Measuring the angle between the extended side and the same side it came from, instead of the angle with the next side.

Exterior Angles Make One Turn

  • For any convex polygon, taking one exterior angle at each vertex, the exterior angles add up to one complete turn.
  • This total stays the same for any convex polygon, whatever its number of sides.
  • Proof idea: there are n linear pairs, and we subtract the interior angle sum. Or walk around and turn once.
Sum of interior angles (n sides)
\[(n-2) \times 180^\circ\]
Exterior sum from linear pairs
\[n \times 180^\circ - (n-2) \times 180^\circ = 2 \times 180^\circ\]
Sum of exterior angles
\[e_1 + e_2 + \cdots + e_n = 2 \times 180^\circ\]

Common mistake: Thinking the exterior angle sum grows with more sides. Only the interior angle sum depends on n.

Each Exterior Angle: Regular Polygon

  • In a regular polygon, all exterior angles are equal.
  • So divide the total of one full turn equally among the n vertices.
Each exterior angle
\[e = \frac{2 \times 180^\circ}{n}\]
Number of sides from e
\[n = \frac{2 \times 180^\circ}{e}\]
Each interior angle
\[i = 180^\circ - \frac{2 \times 180^\circ}{n}\]

Common mistake: Dividing the interior angle sum by n and calling it the exterior angle, or dividing by the wrong number of sides.

Finding Unknown Exterior Angles

  • If some exterior angles are known, subtract their sum from one full turn to get the missing angle or angles.
  • For a ratio a:b:c:d, write the angles as ak, bk, ck, dk and set their sum equal to one full turn.
Missing angle
\[e_{\text{missing}} = 2 \times 180^\circ - (\text{sum of known exterior angles})\]
Ratio method
\[k(a+b+c+d) = 2 \times 180^\circ\]
Equal unknown angles
\[m\,y = 2 \times 180^\circ - (\text{sum of known angles})\]

Common mistake: Finding k in the ratio method but forgetting to multiply it by the ratio part that the question asks for.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. What is the sum of the measures of the exterior angles of a quadrilateral?

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    Answer: D. 360 degrees

    The sum of the measures of the exterior angles of any polygon, including a quadrilateral, is always 360 degrees.

  2. Two exterior angles of a quadrilateral measure 75 degrees and 100 degrees respectively. If the other two exterior angles are equal, what is their measure?

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    Answer: A. 92.5 degrees

    Let x be the measure of each of the equal angles. Then 75 + 100 + 2x = 360. Solving for x: 2x = 185, x = 92.5 degrees.

  3. Prove that the sum of the exterior angles of a convex polygon is 360 degrees using the concept of the sum of interior angles.

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    Answer: C. Consider the sum of interior and exterior angles at each vertex

    At each vertex, the sum of the interior and exterior angles is 180 degrees. The sum of all interior angles is (n-2)180 degrees where n is the number of sides. Subtracting this from the sum of interior and exterior angles proves the theorem.

  4. A regular octagon has exterior angles of equal measure. What is the measure of each exterior angle?

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    Answer: A. 45 degrees

    360/8 = 45

  5. If the ratio of the exterior angles of a quadrilateral is 1:2:3:4, what is the measure of the smallest exterior angle?

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    Answer: A. 36 degrees

    10x = 360; x = 36

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. A pentagon has exterior angles measuring 60, 70, 80, and 90 degrees. What is the measure of the fifth exterior angle?
  2. A regular hexagon has exterior angles all of equal measure. What is the measure of each exterior angle?
  3. Explain why the sum of the exterior angles of any polygon is always 360 degrees.
  4. A quadrilateral has exterior angles in the ratio 2:3:4:6. Find the measure of the largest exterior angle.
  5. A non-convex polygon can have exterior angles whose sum is greater than 360 degrees. Explain why this is possible and provide a simple example.
  6. What is the sum of the exterior angles of a triangle?
  7. If the exterior angles of a quadrilateral are x, 2x, 3x, and 4x, find the value of x.
  8. Three exterior angles of a quadrilateral are 60, 70, and 80 degrees. What is the measure of the fourth exterior angle?
  9. Can a quadrilateral have exterior angles of 90, 90, 90, and 90 degrees? Explain.
  10. What is the relationship between the sum of interior angles and the sum of exterior angles of a polygon?
  11. Explain how to find the measure of an exterior angle of a regular polygon given the number of sides.
  12. Prove that the sum of exterior angles of a convex polygon is 360 degrees using a geometrical method.

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