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? m∠x = m∠y = m∠z = type the correct answer in each box. use numerals instead of words. for help, see this worked example.

Explanation:

Step1: Recall property of cyclic quadrilateral

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

Step2: Find $\angle Y$

Since $\angle W = 68^{\circ}$ and $\angle W+\angle Y=180^{\circ}$, then $\angle Y=180 - 68=112^{\circ}$.

Step3: Solve for $x$ using $\angle X$ and $\angle Z$ relation

We know that $\angle X=(3x + 48)^{\circ}$ and $\angle Z=(5x - 60)^{\circ}$, and $\angle X+\angle Z = 180^{\circ}$. So, $(3x + 48)+(5x - 60)=180$.
Combining like - terms: $3x+5x+48 - 60=180$, which simplifies to $8x-12 = 180$.
Adding 12 to both sides: $8x=180 + 12=192$.
Dividing both sides by 8: $x=\frac{192}{8}=24$.

Step4: Find $\angle X$

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

Step5: Find $\angle Z$

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

Answer:

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