Mathematics: Find ∠MPN Help me please

Maria first made an error in step 3. Maria can correct her error by multiplying 136.49 by 1 over 4 before subtracting 3.0625 from it....

Find ∠MPN Help me please - id13907930, ksawyer745

Ответ:

tiana02q

07.08.2021

Mathematics

$22^{\circ}$

Step-by-step explanation:

Line $\overline{PM}$ is a diameter of the circle because it passes through the circle's center O. Therefore, arc $\widehat{PLM}$ must be 180 degrees, as these are 360 degree in a circle.

We can then find the measure of arc $\widehat{LM}$ :

$\widehat{LP}+\widehat{LM}=180^{\circ},\\92^{\circ}+\widehat{LM}=180^{\circ},\\\widehat{LM}=88^{\circ}$

Arc $\widehat{LM}$ is formed by angle $\angle LPM$ . Define an inscribed angle by an angle with a point on the circle creating an arc on the circumference of the circle. The measure of an inscribed angle is exactly half of the measure of the arc it forms.

Therefore, the measure of $\angle LPM$ must be:

$m\angle LPM=\frac{88}{2}=44^{\circ}$

Similarly, the measure of $\angle LNP$ must be:

$m\angle LNP=\frac{92}{2}=46^{\circ}$

Angles $\angle LPM$ and $\angle MPN$ form angle $\angle LPN$ , which is one of the three angles in $\triangle LPN$ . Since the sum of the interior angles of a triangle add up to 180 degrees, we have:

$(\angle MPN+\angle LPM)+\angl+ PLN+\angle LNP=180^{\circ},\\\angle MPN+44+46+68=180,\\\angle MPN=180-44-46-68,\\\angle MPN=\boxed{22^{\circ}}$

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