Question

pq is tangent to ⊙n at point m. determine m∠p and m∠q. enter your answers in the boxes. m∠p = m∠q =

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle NMP = 90^{\circ}$ and $\angle NMQ=90^{\circ}$.

Step2: Find $m\angle P$ in $\triangle NMP$

In $\triangle NMP$, we know that the sum of the interior angles of a triangle is $180^{\circ}$. Given $\angle MNP = 64^{\circ}$ and $\angle NMP = 90^{\circ}$. Let $m\angle P=x$. Then $x + 64^{\circ}+90^{\circ}=180^{\circ}$. Solving for $x$: $x=180^{\circ}-(90^{\circ} + 64^{\circ})=26^{\circ}$.

Step3: Find $m\angle Q$ in $\triangle NMQ$

In $\triangle NMQ$, we know that the sum of the interior angles of a triangle is $180^{\circ}$. Given $\angle MNQ = 36^{\circ}$ and $\angle NMQ = 90^{\circ}$. Let $m\angle Q = y$. Then $y+36^{\circ}+90^{\circ}=180^{\circ}$. Solving for $y$: $y = 180^{\circ}-(90^{\circ}+36^{\circ}) = 54^{\circ}$.

Answer:

$m\angle P = 26^{\circ}$
$m\angle Q = 54^{\circ}$