Question

Hi can you help me? I am struggling with the formula

Answer 1

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:

`\begin{gathered} \angle2=(arcCE-arcBD)/(2) \\ \\ \angle1=(arcCE+arcBD)/(2) \end{gathered}`

We now have the two equations.

Thus, we have:

Let x represent arcCE

Let y represent arc BD.

Thus, we have:

`\begin{gathered} 31=(x-y)/(2) \\ \\ 78=(x+y)/(2) \end{gathered}`

Let's solve the system of equations simultaneously.

Rewrite the first equation for x:

`\begin{gathered} (x-y)/(2)=31 \\ \\ x-y=31(2) \\ \\ x-y=62 \\ \\ x=62+y \end{gathered}`

Substitute in (62+y) for x in the second equation:

`\begin{gathered} 78=(62+y+y)/(2) \\ \\ 78=(62+2y)/(2) \\ \\ 78(2)=62+2y \\ \\ 156=62+2y \\ \\ \text{ Subtract 62 from both sides:} \\ 156-62=62-62+2y \\ \\ 94=2y \\ \\ (94)/(2)=y \\ \\ 47=y \\ \\ y=47^o \end{gathered}`

Now, plug in 47 for y in either of the equations:

`\begin{gathered} x=62+y \\ \\ x=62+47 \\ \\ x=109^o \end{gathered}`

• 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°