Chapter 9.4, Problem 2CE

To determine

To Prove: The arc PQ of circle M is congruent to arc RS of the circle N.

Explanation of Solution

Given information:

  

In given figure two circles contain chords $PQ,XY,RS$ and point $M$ as a center point of first circle and point N as center point of second circle.

In figure ΔPMQ and ΔSNR

Chords are congruent, $PQ\cong RS$

Also their two sides

  $MQ\cong NR$

And

  $MP\cong NS$

Because they are not radius of same circle.

And not any angle $\angle PMQ\cong \angle SNR$ because they are not vertical opposite angles.

Hence, two triangles ΔPMQ and ΔSNR does not satisfy SAS congruency or any other congruency property.

Then it concludes that $\Delta PMQ\ne \Delta SNR$

If $\Delta PMQ\ne \Delta SNR$ then its corresponding parts such as arcs $PQ\ne RS$

Therefore, it proves that arc $PQ\ne RS.$