Question

What is the measure of arc AB when m angle APB=78 102 156 204 39

Answer 1

In the given figure above, we can use the formula for tangent-tangent angle. This is under the type of angle whose vertex is outside the circle and its sides intersects the circle. 

Applying the tangent-tangent angle formula: 

m∠APB = $\frac{1}{2}$ (ALB - AB)

m∠APB = angle of the vertex
ALB = measure of arc ALB
AB = measure of arc AB

we all know that the total measure of arc is equal to 360, hence, ALB + AB = 360

let:  x = measure of arc AB
      360 - x = measure of arc ALB 

substitute: 

78 = $\frac{1}{2}$ {(360 - x) - x}
78 = $\frac{1}{2}$ (360 - 2x)
78 = 180 - x     ⇒ simplifying further by dividing both 360 and -2x by 2
x = 180 - 78     ⇒ combining like terms
x = 102     
arc AB = 102   ⇒ Answer