Question

a quadrilateral has two angles that measure 186° and 72°. the other two angles are in a ratio of 6:11. what are the measures of those two angles?

Explanation:

Step1: Find the sum of the four - angle measures of a quadrilateral

The sum of the interior angles of a quadrilateral is $(4 - 2)\times180^{\circ}=360^{\circ}$.

Step2: Calculate the sum of the two unknown angles

Let the sum of the two unknown angles be $x$. We know two angles are $186^{\circ}$ and $72^{\circ}$. So $x=360-(186 + 72)=360 - 258=102^{\circ}$.

Step3: Use the ratio to find the individual angles

The two angles are in a ratio of $6:11$. Let the common multiplier be $k$. Then the two angles are $6k$ and $11k$. And $6k + 11k=102$, so $17k = 102$. Solving for $k$, we get $k=\frac{102}{17}=6$.
The first angle is $6k = 6\times6 = 36^{\circ}$, and the second angle is $11k=11\times6 = 66^{\circ}$.

Answer:

$36$ and $66$