Is Converse True ?

[goelmohit2002]

Hi All,

Angle property for a circle is that

"Angle in a semi circle is 90 degrees"

Can someone please tell is the reverse also true...i.e.

"If a angle in a circle is given to be 90 degrees...then is it necessary that angle is inscribed in a semi circle"....if yes, then can someone please explain why so ?

Thanks
Mohit

[Ben Ku]

Hi Mohit,

Yes, the converse of the theorem is also true:

If an angle inscribed in a circle is 90 degrees, then its intercepted arc is a semi-circle.

To prove it, we need to know this theorem:
The measure of an inscribed angle is half the measure of the intercepted arc.

For example, if an angle inscribed in a circle cuts off an arc of 80 degrees, then that angle has a measure of 40 degrees. The converse of this theorem is also true.

Because an inscribed right angle will always cut off an arc that measures 180 degrees, it always cuts off a semi-circle.

Let me know if that makes sense!

[goelmohit2002]

Hi Mohit,

Yes, the converse of the theorem is also true:

If an angle inscribed in a circle is 90 degrees, then its intercepted arc is a semi-circle.

To prove it, we need to know this theorem:
The measure of an inscribed angle is half the measure of the intercepted arc.

For example, if an angle inscribed in a circle cuts off an arc of 80 degrees, then that angle has a measure of 40 degrees. The converse of this theorem is also true.

Because an inscribed right angle will always cut off an arc that measures 180 degrees, it always cuts off a semi-circle.

Let me know if that makes sense!

Thanks Ben Ku !

[Ben Ku]

You're welcome!