Question

in quadrilateral wxyz m∠w is 68°, m∠x is (3x + 48)°, and m∠z is (5x - 60)°. what are the measures of ∠x, ∠y, and ∠z? type the correct answer in each box. use numerals instead of words. for help, see this worked - sample. m∠x = m∠y = m∠z =

Explanation:

Step1: Recall property of cyclic quadrilateral

In a cyclic quadrilateral, the sum of opposite angles is 180°. So, $\angle W$ and $\angle Y$ are opposite, and $\angle X$ and $\angle Z$ are opposite. So, $m\angle X + m\angle Z=180^{\circ}$.

Step2: Set up equation

Set up the equation $(3x + 48)+(5x - 60)=180$.
Combine like - terms: $3x+5x+48 - 60=180$, which simplifies to $8x-12 = 180$.
Add 12 to both sides: $8x=180 + 12=192$.
Divide both sides by 8: $x=\frac{192}{8}=24$.

Step3: Find $m\angle X$

Substitute $x = 24$ into the expression for $m\angle X$.
$m\angle X=3x + 48=3\times24+48=72 + 48=120^{\circ}$.

Step4: Find $m\angle Z$

Substitute $x = 24$ into the expression for $m\angle Z$.
$m\angle Z=5x - 60=5\times24-60=120 - 60 = 60^{\circ}$.

Step5: Find $m\angle Y$

Since $m\angle W=68^{\circ}$ and $\angle W$ and $\angle Y$ are opposite in the cyclic quadrilateral, $m\angle Y=180 - 68=112^{\circ}$.

Answer:

$m\angle X = 120$
$m\angle Y = 112$
$m\angle Z = 60$