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Apply the exterior angle-sum relationship for polygons

For a convex polygon, the exterior angles formed by extending one side at each vertex and measuring the consistent turning direction have a sum of 360∘360^\circ, because the sides make one complete revolution. This relationship, together with the supplementary relationship between an interior angle and its adjacent exterior angle, supports finding unknown angles and, for regular polygons, determining each exterior angle or the number of sides; nonconvex and more advanced generalized cases are not included.

Detailed Explanation: Apply the exterior angle-sum relationship for polygons

For any convex polygon, the exterior angles are the turns you make as you travel around the polygon in the same direction. One complete turn is 360∘360^\circ, so:

sum of exterior angles=360∘\text{sum of exterior angles}=360^\circ

If an exterior angle and its adjacent interior angle form a straight line, they are supplementary:

interior angle+exterior angle=180∘\text{interior angle}+\text{exterior angle}=180^\circ

Example: A convex pentagon has exterior angles of 65∘65^\circ, 72∘72^\circ, 80∘80^\circ, 68∘68^\circ, and x∘x^\circ. Find xx and the interior angle next to it.

Step 1: Use the exterior-angle sum.

65+72+80+68+x=36065+72+80+68+x=360

Step 2: Add the known exterior angles.

285+x=360285+x=360

Step 3: Solve for xx.

x=360−285=75∘x=360-285=75^\circ

So, the missing exterior angle is:

75∘\boxed{75^\circ}

Step 4: Find the adjacent interior angle, if needed.

The interior angle and exterior angle add to 180∘180^\circ:

interior angle+75=180\text{interior angle}+75=180 interior angle=105∘\text{interior angle}=105^\circ

Therefore, the adjacent interior angle is:

105∘\boxed{105^\circ}

Always make sure the exterior angles are measured as turns in the same direction around the convex polygon.

Learn by doing: Apply the exterior angle-sum relationship for polygons

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