Question

two chords intersect in a circle forming right angles. what is the sum of the measures of a pair of opposite arcs

Answer 1

If two chords, say AB and then BC intersect at B so that ABC is 90 degrees,
then you have a right triangle ABC.
Let O be the center of the circle and consider the arc AOC.
By the inscribed angle theorem, ½ Angle (AOC) = Angle (ABC).

But this means Angle (AOC) is 180 degrees.
So A-O-C is a straight line segment.
But O is the center, so AC must then be the diameter.

So the line segment AC splits the circle into two equal arcs, both of 180 degrees.
So Arc AB + Arc BC = 180.

Answer 2

thanks

Answer 3

no problem

Answer 4

dont understand