NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions
Question 78

Question. 78

Draw an angle of 65° and draw an angle equal to this angle, using ruler and compasses.

Answer:

Construction: (1) Draw ∠ABC = 65° (protractor). (2) At point P elsewhere, draw a ray PQ. (3) With centre B take arc cutting BA and BC at E and F; copy the same radius at P to cut PQ at R. (4) With centres E and F distances, reproduce intersection on PR to get point S; join PS to form ∠QPS equal to 65°.

Detailed Answer with Explanation:

78. Draw an angle of 65° and then draw another angle equal to it (using ruler and compasses only).

Simple Tools Needed

  • Ruler (without markings for measuring angles)
  • Compass
  • Pencil
  • Protractor (only to first make the given 65° angle)

Step A: First, draw the given 65° angle

  1. Draw a point B on your page. Draw a ray BA.
  2. Place the protractor at point B on ray BA and mark (65^circ).
  3. Through that mark, draw the second ray BC.

You now have angle (angle ABC) equal to (65^circ).

Step B: Start a new place to copy the same angle

  1. Pick any new point P elsewhere on the page.
  2. Draw a ray PQ starting at P.
  3. We will now copy the size of (angle ABC) onto point P using the compass.

Step C: Make a reference arc on the original angle

  1. Place the compass needle at B. Open the compass to any comfortable width.
  2. Draw an arc that cuts both rays BA and BC.
  3. Mark the cut points: on BA name it E; on BC name it F.

So, (BE = BF) because both are radii of the same arc.

Step D: Copy that same arc at point P

  1. Without changing the compass width, move the compass to point P.
  2. Draw a similar arc that cuts the ray PQ. Mark this cut point as R.

Now (PR) has the same length as (BE) and (BF) from the original step.

Step E: Copy the “chord” length between the two arc points

  1. Go back to the original angle at B.
  2. Set the compass width to the distance (EF) (the straight distance between the two arc marks on BA and BC).
  3. Return to point P. With the compass needle at R, draw a small arc crossing the big arc you made in Step D.
  4. Mark this new intersection as S.

Step F: Complete the copied angle

  1. Draw the ray PS.
  2. The angle at P between rays PQ and PS is your copied angle.

Therefore, (angle QPS = angle ABC) and its size is (65^circ).

Why this works (very simple reason)

  • In the original angle, the arc from E to F has a certain “chord length” (EF).
  • We copied the same arc radius and the same chord length at point P.
  • When two arcs have the same radius and the same chord, they cut out equal angles at the center.

Hence, the new angle at P is exactly equal to the given (65^circ) angle.

Very Short Recap

  1. Make (angle ABC = 65^circ) with a protractor.
  2. Draw a new ray PQ.
  3. Copy the arc (same radius) → mark R on PQ.
  4. Copy the chord (EF) to get point S on the copied arc.
  5. Join PS. Then (angle QPS = 65^circ).
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NCERT Exemplar Solutions Class 6 – Mathematics – Unit 9: Symmetry and Practical Geometry – Problems and Solutions | Detailed Answers