The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.
Thus,
• Angle F is ,half, of Arc EHG
,
• Angle H is ,half, of Arc EFG
Together, Arc EHG and Arc EFG makes up a circle. This is equal to 360°.
We can write:
[tex]\begin{gathered} \angle F=\frac{1}{2}\text{ArcEHG} \\ \angle H=\frac{1}{2}\text{ArcEFG} \\ \angle F+\angle H=\frac{1}{2}\text{ArcEHG}+\frac{1}{2}\text{ArcEFG} \\ \angle F+\angle H=\frac{1}{2}(ArcEHG+ArcEFG) \\ \angle F+\angle H=\frac{1}{2}(360) \\ \therefore\angle F+\angle H=180 \end{gathered}[/tex]
Now, we can substitute the expressions for Angle F and Angle H and solve for x. Shown below:
[tex]\begin{gathered} \angle F+\angle H=180 \\ (6x-26)+(x+45)=180 \\ 6x-26+x+45=180 \\ 7x+19=180 \\ 7x=161 \\ x=23 \end{gathered}[/tex]
