NCERT Exemplar Solutions Class 6 – Mathematics – Unit 9: Symmetry and Practical Geometry – Problems and Solutions

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

Question 83

Question. 83

Bisect ∠XYZ of Fig. 9.22

Construction: Draw arcs from X and Z cutting each other at P. Join YP. Then ∠XYZ is bisected into two equal parts.

Put the compass pin at point Y. Open the compass to a small, comfortable width (any convenient radius). Draw one arc that cuts both arms YX and YZ. Mark the cut points as A on YX and B on YZ.
(A in YX,quad B in YZ)
Without changing the compass width, place the pin on A and draw a small arc in the interior of the angle.
Again, without changing the width, place the pin on B and draw another small arc so that it meets the previous arc. Mark their intersection as P.
Use the ruler to draw a straight line from Y to P. The line YP is the bisector of the angle.
(YP ext{ bisects } angle XYZ)

We drew equal arcs with the same compass width. So the distances are equal.

(YA = YB)

(AP = BP)

Now look at triangles △YAP and △YBP:

So the two triangles are congruent by SSS.

( riangle YAP cong riangle YBP)

Therefore the angles at Y on both sides of YP are equal.

(angle AYP = angle PYB)

This means YP divides the original angle into two equal parts.

(angle XYZ = angle AYP + angle PYB)

(angle AYP = angle PYB)

Use a protractor to measure
(angle AYP)
and
(angle PYB)
. They should be equal.
Or, keep the compass width the same and compare the arcs; both sides should match.

Line YP is the bisector of

(angle XYZ)

, so the angle is split into two equal angles.