Question

for which convex polygon is the sum of the exterior angles equal to 360°? select all that apply.
a triangle
b hexagon
c pentagon
d quadrilateral

Explanation:

The sum of the exterior angles of any convex polygon (one with all interior angles less than \(180^\circ\)) is always \(360^\circ\), regardless of the number of sides. Let's verify for each option:

Step 1: Analyze Triangle (A)

A triangle is a convex polygon (for non - degenerate triangles). The sum of its exterior angles: For a triangle, each exterior angle and its corresponding interior angle are supplementary (\(180^\circ\)). Let the interior angles be \(A\), \(B\), \(C\) with \(A + B + C=180^\circ\). The exterior angles are \(180 - A\), \(180 - B\), \(180 - C\). Summing them: \((180 - A)+(180 - B)+(180 - C)=540-(A + B + C)=540 - 180 = 360^\circ\).

Step 2: Analyze Hexagon (B)

A convex hexagon has 6 sides. For any convex polygon with \(n\) sides, the sum of exterior angles is \(360^\circ\). We can also derive it: Each interior angle \(I\) and exterior angle \(E\) satisfy \(I+E = 180^\circ\). The sum of interior angles of a polygon is \((n - 2)\times180^\circ\). So sum of exterior angles \(=\sum_{i = 1}^{n}E_i=\sum_{i = 1}^{n}(180 - I_i)=n\times180-\sum_{i = 1}^{n}I_i=n\times180-(n - 2)\times180=360^\circ\). For \(n = 6\) (hexagon), this holds.

Step 3: Analyze Pentagon (C)

A convex pentagon has \(n = 5\) sides. Using the same formula as above, sum of exterior angles \(=n\times180-(n - 2)\times180=360^\circ\). So for a pentagon, the sum of exterior angles is \(360^\circ\).

Step 4: Analyze Quadrilateral (D)

A convex quadrilateral has \(n=4\) sides. Using the formula for sum of exterior angles of a polygon, we get \(4\times180-(4 - 2)\times180=720 - 360=360^\circ\). So the sum of exterior angles of a quadrilateral is \(360^\circ\).

Answer:

A. Triangle, B. Hexagon, C. Pentagon, D. Quadrilateral