PLEASE HELP!!!Consider circle O, where m BD=70 and m CA=170

Accepted Solution

Answer:1. [tex]m\angle BPD=120^{o}[/tex]2. [tex]m BC+m AD=120^{o}[/tex] Step-by-step explanation: We have been given a circle O, where measure of arc BD is 70 degrees and measure of arc CA is 170 degrees.To find the measure of angle BPD we will use intersecting chords theorem. This theorem states that the measure of an angle formed by two  chords that intersect within a circle is 1/2 the sum of the measures of the arcs  intercepted by the angle and its vertical angle.We can see from our diagram that arc BD intercepts angle BPD and arc CA intercepts angle CPA. [tex]m\angle BPD=\frac{1}{2}(m CA+m BD)[/tex][tex]m\angle BPD=\frac{1}{2}(170+70)[/tex][tex]m\angle BPD=\frac{1}{2}(240)[/tex][tex]m\angle BPD=120[/tex] Therefore, measure of angle BPD is 120 degrees.  Since we know that the arc measure can also be find by measure of a central angle of circle. Since 2πR is the circumference of the whole circle, so it is same as the ratio of the arc angle to a full angle (360).  Let us find measure of arc BC and AD by subtracting the measures of arc BD and CA from full angle of a circle.   [tex]m BC+m AD=360-(m BD+m CA)[/tex] [tex]m BC+m AD=360-(70+170)[/tex] [tex]m BC+m AD=360-(240)[/tex] [tex]m BC+m AD=120[/tex] Therefore, the measure of arc AD and arc BC is 120 degrees.