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Class 9 Maths — Proving angle facts

Use the linear-pair axiom to prove vertically opposite angles equal, and apply the angle-sum and exterior-angle theorems.

Name: ________________________

Before you start

One side of this triangle has been carried straight on past the corner B. Find out what each of the four angles here is called.

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  1. 1. The interior angle at A
  2. 2. The interior angle at B
  3. 3. The interior angle at C
  4. 4. The exterior angle at B

The exterior angle at B equals the two interior angles it does not touch — the ones at A and C — added together. That is the exterior angle theorem, and it follows straight from the angle sum.

  1. 1.Sort each set of angles by whether they must be equal or must add up to 180°.

    Groups: Must be equal · Must add up to 180°

    • A linear pair
    • Alternate interior angles on parallel lines
    • Co-interior angles on parallel lines
    • Corresponding angles on parallel lines
    • Vertically opposite angles
    • The three angles of a triangle

  2. 2.The three angles of a triangle are in the ratio 1 : 2 : 3. How big is the largest, in degrees?

  3. 3.Parallel lines are cut by a transversal. Two co-interior angles measure 2x degrees and x degrees. What is x?

  4. 4.Match each description to the name of the angle pair.

    • Same side of the transversal, one inside the parallel lines and one outside
    • Opposite sides of the transversal, both inside the parallel lines
    • Same side of the transversal, both inside the parallel lines
    • Facing each other where two straight lines cross
    • Alternate interior angles
    • Co-interior angles
    • Corresponding angles
    • Vertically opposite angles

  5. 5.Parallel lines are cut by a transversal, and one of a pair of alternate interior angles is 73°. What is the other, in degrees?

  6. 6.Two angles form a linear pair. One of them is 118°. What is the other, in degrees?

  7. 7.The proof that vertically opposite angles are equal reaches the line a + b = b + c. What is taken from both sides next?

    • a) b
    • b) c
    • c) 180
    • d) a

  8. 8.Two straight lines cross. One of the four angles is 47°. What is the angle vertically opposite it, in degrees?

Answer key — Class 9 Maths — Proving angle facts

  1. 1. Vertically opposite angles → Must be equal; A linear pair → Must add up to 180°; Alternate interior angles on parallel lines → Must be equal; Co-interior angles on parallel lines → Must add up to 180°; Corresponding angles on parallel lines → Must be equal; The three angles of a triangle → Must add up to 180°
  2. 2. 90
  3. 3. 60
  4. 4. Same side of the transversal, one inside the parallel lines and one outside → Corresponding angles; Opposite sides of the transversal, both inside the parallel lines → Alternate interior angles; Same side of the transversal, both inside the parallel lines → Co-interior angles; Facing each other where two straight lines cross → Vertically opposite angles
  5. 5. 73
  6. 6. 62
  7. 7. a) b
  8. 8. 47