Question

find the measure of the arc or central angle indicated. assume that lines which appear to be diameters are actual diameters.

1. ( moverarc{str} )
2. ( mangle igj )
3. ( mangle fdg )
4. ( mangle igj )

solve for ( x ). assume that lines which appear to be diameters are actual diameters.
5)
6)

Explanation:

Step1: Calculate arc $\overparen{STR}$

The total circle is $360^\circ$. Subtract the given arc $138^\circ$.
$m\overparen{STR} = 360^\circ - 138^\circ = 222^\circ$

Step2: Calculate $\angle IGJ$

Total circle is $360^\circ$. Subtract the two given arcs.
$m\angle IGJ = 360^\circ - 150^\circ - 60^\circ = 150^\circ$

Step3: Calculate $\angle FDG$

Total circle is $360^\circ$. Subtract the three given arcs.
$m\angle FDG = 360^\circ - 155^\circ - 50^\circ - 90^\circ = 65^\circ$

Step4: Calculate $\angle IGJ$

$\angle IGJ$ is supplementary to $75^\circ$ (since $HJ$ is a diameter, $180^\circ$ total).
$m\angle IGJ = 180^\circ - 75^\circ = 105^\circ$

Step5: Solve for $x$ (Problem 5)

Sum of arcs equals $360^\circ$. Set up the equation.
$117^\circ + 118^\circ + (38x + 4)^\circ = 360^\circ$
Simplify: $239 + 38x = 360$
$38x = 360 - 239 = 121$
$x = \frac{121}{38} \approx 3.18$

Step6: Solve for $x$ (Problem 6)

Sum of arcs equals $360^\circ$. Set up the equation.
$(7x + 1)^\circ + 42^\circ + 88^\circ + 2\times(7x + 1)^\circ = 360^\circ$
Simplify: $3(7x+1) + 130 = 360$
$21x + 3 + 130 = 360$
$21x = 360 - 133 = 227$
$x = \frac{227}{21} \approx 10.81$

Answer:

1. $m\overparen{STR} = 222^\circ$
2. $m\angle IGJ = 150^\circ$
3. $m\angle FDG = 65^\circ$
4. $m\angle IGJ = 105^\circ$
5. $x = \frac{121}{38}$ or $\approx 3.18$
6. $x = \frac{227}{21}$ or $\approx 10.81$