Question

challenge given that m∠a=(18x)°, m∠c=(8x + 23)°, and m∠d=113°, what is m∠b? m∠b=□°

Explanation:

Step1: Recall sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. So, $m\angle A + m\angle B+m\angle C + m\angle D=360^{\circ}$.

Step2: Substitute given angle - measures

We know that $m\angle A=(18x)^{\circ}$, $m\angle C=(8x + 23)^{\circ}$, and $m\angle D = 113^{\circ}$. Substituting these into the equation gives us $(18x)+m\angle B+(8x + 23)+113 = 360$.

Step3: Combine like - terms

Combine the $x$ terms and the constant terms: $(18x+8x)+m\angle B+(23 + 113)=360$, which simplifies to $26x+m\angle B + 136=360$.

Step4: Isolate the terms with $x$ and $m\angle B$

Subtract 136 from both sides of the equation: $26x+m\angle B=360 - 136$, so $26x+m\angle B=224$.

Step5: Since we assume this is a cyclic quadrilateral (if no other information, and using the property of angles in a quadrilateral), we can also use the fact that opposite angles are supplementary in a cyclic quadrilateral. But without more context, we'll solve for $x$ first from the sum - of - angles formula.

We know that $26x+m\angle B=224$. We need to find $x$. Since we have no other relation, we assume this is a general quadrilateral.
We rewrite the sum of angles equation as $18x+8x+23 + 113+m\angle B=360$.
Combining like terms: $26x+136+m\angle B=360$.
$26x+m\angle B=224$.
If we assume this is a cyclic quadrilateral, $\angle A+\angle C = 180^{\circ}$ (opposite angles are supplementary).
So, $18x+8x + 23=180$.
Combining like terms: $26x=180 - 23$, $26x=157$, $x=\frac{157}{26}\approx6.04$.
Substitute $x$ into the sum of angles equation:
$m\angle A=18x=18\times\frac{157}{26}=\frac{2826}{26}\approx108.69^{\circ}$, $m\angle C=8x + 23=8\times\frac{157}{26}+23=\frac{1256}{26}+23=\frac{1256+598}{26}=\frac{1854}{26}\approx71.31^{\circ}$, $m\angle D = 113^{\circ}$.
Since $m\angle A + m\angle B+m\angle C + m\angle D=360^{\circ}$, then $m\angle B=360-(m\angle A + m\angle C + m\angle D)$.
$m\angle A + m\angle C + m\angle D=108.69+71.31+113=293^{\circ}$.

Answer:

$m\angle B = 67^{\circ}$