Angle abd measures (4x + 10)^o. angle acd measures (5x − 2)^o what is the measure of arc ad?

1. check the picture at the bottom of the solution.

2. abd and acd are inscribed angles(angles formed by 3 points in a circle) intercepting the same arc, ad, so by an well-known property, their measures are equal.

3. 5x-2°=4x+10°
          x=12°

so to find the measure of the angle abd(=m(acd)) substitute x in any of 5x-2 or 4x+10. in both cases we get m(abd)=58°

4. the measure of the arc ad is twice the measure of the inscribed arc abd, so the measure of arc ad is 116°