Given:
• m∠1 = 78 degrees
,
• m∠2 = 31 degrees.
Let's solve for the following:
• Measure of arc CE:
To find the measure of arcCE, apply the angle-arc relationship:
[tex]\begin{gathered} \angle2=\frac{arcCE-arcBD}{2} \\ \\ \angle1=\frac{arcCE+arcBD}{2} \end{gathered}[/tex]
We now have the two equations.
Thus, we have:
Let x represent arcCE
Let y represent arc BD.
Thus, we have:
[tex]\begin{gathered} 31=\frac{x-y}{2} \\ \\ 78=\frac{x+y}{2} \end{gathered}[/tex]
Let's solve the system of equations simultaneously.
Rewrite the first equation for x:
[tex]\begin{gathered} \frac{x-y}{2}=31 \\ \\ x-y=31(2) \\ \\ x-y=62 \\ \\ x=62+y \end{gathered}[/tex]
Substitute in (62+y) for x in the second equation:
[tex]\begin{gathered} 78=\frac{62+y+y}{2} \\ \\ 78=\frac{62+2y}{2} \\ \\ 78(2)=62+2y \\ \\ 156=62+2y \\ \\ \text{ Subtract 62 from both sides:} \\ 156-62=62-62+2y \\ \\ 94=2y \\ \\ \frac{94}{2}=y \\ \\ 47=y \\ \\ y=47^o \end{gathered}[/tex]
Now, plug in 47 for y in either of the equations:
[tex]\begin{gathered} x=62+y \\ \\ x=62+47 \\ \\ x=109^o \end{gathered}[/tex]
• Since x represents the measure of arcCE,:
Measure of arcCE = 109°
• Since y represents the measure of arcBD:
Measure of arcBD = 47°
ANSWER:
• Measure of arcCE = 109°
,
• Measure of arcBD = 47°
