Question

12 fill in the blank 3 points
given $overline{il}$ is a diameter. $mangle icr=(3x + 5)^{circ}$ & $mangle rcl=(x - 1)^{circ}$
$x=$ type your answer...
$mangle icr=$ type your answer...
what is the measure of arc ilr? type your answer... $^{circ}$

Explanation:

Step1: Set sum to 180° (straight angle)

Since $\overline{IL}$ is a diameter, $\angle ICR + \angle RCL = 180^\circ$. Substitute the given expressions:
$$(3x + 5) + (x - 1) = 180$$

Step2: Simplify and solve for $x$

Combine like terms and isolate $x$:
$$4x + 4 = 180$$
$$4x = 176$$
$$x = \frac{176}{4} = 44$$

Step3: Calculate $m\angle ICR$

Substitute $x=44$ into $(3x + 5)$:
$$m\angle ICR = 3(44) + 5 = 132 + 5 = 137$$

Step4: Find measure of arc $ILR$

The measure of an arc equals the measure of its central angle. Arc $ILR$ is the major arc, so its measure is $360^\circ - m\angle ICL$. Since $\angle ICL$ is a straight angle ($180^\circ$), or alternatively, arc $ILR$ corresponds to the reflex angle at $C$ for arc $ILR$:
$$m\overarc{ILR} = 360^\circ - m\overarc{IR}$$
Since $m\overarc{IR} = m\angle ICR = 137^\circ$,
$$m\overarc{ILR} = 360 - 137 = 223$$

Answer:

$x=44$
$m\angle ICR=137^\circ$
The measure of arc $ILR$ is $223^\circ$