Question

if, for a convex octagon, five exterior angles each measure 38° and two exterior angles each measure 29°, find the measure of the eighth exterior angle. 119° 115° 116° 112° none of these answers are correct. question 30 1 p if three of the interior angles of a convex quadrilateral measure 95°, 116°, and 80°, find the measure of the fourth interior angle. 58° 69° 65° 63° none of these answers are correct.

Explanation:

Step1: Recall the sum of exterior angles of a polygon

The sum of the exterior angles of any convex polygon is \(360^{\circ}\).

Step2: Calculate the sum of the given exterior angles

Five exterior angles each of \(38^{\circ}\) sum to \(5\times38^{\circ}=190^{\circ}\), and two exterior angles each of \(29^{\circ}\) sum to \(2\times29^{\circ} = 58^{\circ}\). The total of the given seven exterior angles is \(190^{\circ}+58^{\circ}=248^{\circ}\).

Step3: Find the eighth exterior angle

Let the eighth exterior angle be \(x\). Using the formula \(x=360^{\circ}-\text{(sum of given exterior angles)}\), we have \(x = 360^{\circ}-248^{\circ}=112^{\circ}\).

Step1: Recall the sum of interior angles of a quadrilateral

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

Step2: Calculate the sum of the given interior angles

The sum of the three given interior angles \(95^{\circ}+116^{\circ}+80^{\circ}=291^{\circ}\).

Step3: Find the fourth interior angle

Let the fourth interior angle be \(y\). Using the formula \(y = 360^{\circ}-\text{(sum of given interior angles)}\), we have \(y=360^{\circ}-291^{\circ}=69^{\circ}\).

Answer:

\(112^{\circ}\)

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