Construct at least 4 different angles in different orientations without taking any measurement. Make a copy of all these angles.
Step 1: Draw the given angle ∠BAC.
Step 2: Draw a ray XY.
Step 3: With A as centre and any radius, draw an arc cutting AB at P and AC at Q.
Step 4: With X as centre and the same radius, draw an arc cutting XY at Z.
Step 5: Measure PQ using compass and cut an arc from Z with radius PQ, intersecting the previous arc at R.
Step 6: Join XR. ∠YXR is a copy of ∠BAC.
Why it works: Triangles APQ and XZR are congruent by SSS, making the angles equal.
Explanation
The chapter describes the method to copy an angle using only ruler and compass. The key insight is that an angle is defined by three points forming an isosceles triangle. By copying the three sides of this triangle (using compass for lengths), we create a congruent triangle, and corresponding angles of congruent triangles are equal. This method works for angles in any orientation.
Solution Steps
Draw the given angle ∠BAC.
Draw a ray XY.
With A as centre and any radius, draw an arc cutting AB at P and AC at Q.
With X as centre and the same radius, draw an arc cutting XY at Z.
Measure PQ using compass and cut an arc from Z with radius PQ, intersecting the previous arc at R.
Join XR to complete the copied angle ∠YXR.