NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions
Question 73

Question. 73

Draw the images P′, Q′ and R′ of the points P, Q and R, respectively in the line n (Fig. 9.18). Join P′Q′ and Q′R′ to form an angle P′Q′R′. Measure ∠PQR and ∠P′Q′R′. Are the two angles equal?

Answer:

Yes, the two angles are equal.

Detailed Answer with Explanation:

Q.73 — Reflection and Angle Measure

Task: Reflect the points P, Q, R in the line n to get P′, Q′, R′. Join P′Q′ and Q′R′ to make angle P′Q′R′. Measure both angles and compare.

Step-by-step Construction

  1. Draw perpendiculars to the mirror line.
    • From point P, draw a line perpendicular to n. Mark its foot as DP.
    • From point Q, draw a line perpendicular to n. Mark its foot as DQ.
    • From point R, draw a line perpendicular to n. Mark its foot as DR.
  2. Mark equal distances on the other side.
    • Measure the distance P DP. Take the same distance on the other side of n along the same perpendicular and mark P′.
    • Repeat for Q to get Q′, and for R to get R′.
  3. Join the reflected segments.
    • Draw the rays/segments P′Q′ and Q′R′.
    • The angle at Q′ formed is ∠P′Q′R′.
  4. Measure the two angles.
    • Use a protractor to measure ∠PQR.
    • Measure ∠P′Q′R′ the same way.

Why the angles are equal (simple idea)

Reflection in a line is a mirror move that keeps shapes the same size and shape — only flipped.

( ext{Reflection is a rigid motion (isometry).} )

( ext{Rigid motions preserve lengths and angles.} )

So when we reflect the rays forming the angle at Q across line n, the new rays at Q′ make the same angle.

Key properties used

  • For any point and its image:

    ( n perp PP' )

    ( D_P ext{ is the midpoint of } PP' )

    ( PD_P = P'D_P )

  • Distances are preserved:

    ( PQ = P'Q' )

    ( QR = Q'R' )

  • Hence, the opening between the reflected rays stays the same:

    ( angle PQR = angle P'Q'R' )

Conclusion

Yes, the two angles are equal.

( oxed{ angle PQR = angle P'Q'R' } )

Reason: Reflection preserves angle measure.

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