Question

find $m\angle cdf$ and $m\angle def$.
$(5x + 2)^\circ$
$(2x + 18)^\circ$
$m\angle cdf = $
number:
$m\angle def = $
color:

Explanation:

Step1: Recognize alternate interior angles

In the parallelogram (since $DE \parallel CF$ and $CD$ is a transversal), $\angle CDF = (5x+2)^\circ$ and $\angle DCF=(2x+18)^\circ$ are alternate interior angles, so they are equal:
$$5x + 2 = 2x + 18$$

Step2: Solve for $x$

Subtract $2x$ and $2$ from both sides:
$$5x - 2x = 18 - 2$$
$$3x = 16$$
$$x = \frac{16}{3} \approx 5.33$$

Step3: Calculate $m\angle CDF$

Substitute $x$ into $(5x+2)$:
$$m\angle CDF = 5\times\frac{16}{3} + 2 = \frac{80}{3} + \frac{6}{3} = \frac{86}{3} \approx 28.67^\circ$$

Step4: Find $m\angle DEF$

In parallelogram $CDEF$, $\angle DEF$ is supplementary to $\angle CDF$ (consecutive angles in a parallelogram sum to $180^\circ$):
$$m\angle DEF = 180 - \frac{86}{3} = \frac{540}{3} - \frac{86}{3} = \frac{454}{3} \approx 151.33^\circ$$

Answer:

$m\angle CDF = \frac{86}{3}^\circ$ (or approximately $28.7^\circ$)
$m\angle DEF = \frac{454}{3}^\circ$ (or approximately $151.3^\circ$)