Question

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the diagram shows a convex polygon.
what is the value of a?
a = \boxed{\space}°

Explanation:

Step1: Recall the sum of exterior angles of a polygon

The sum of the exterior angles of any convex polygon is \( 360^\circ \)? No, wait, for any polygon, the sum of exterior angles (one at each vertex, taken in order) is \( 360^\circ \)? Wait, no, actually, for any convex polygon, the sum of exterior angles (when you extend each side and take the angle outside) is \( 360^\circ \)? Wait, no, that's for regular polygons? Wait, no, the sum of exterior angles of any polygon (convex or not, as long as it's simple) is \( 360^\circ \) when you take one exterior angle at each vertex. Wait, but in this diagram, we have three angles: \( 2a + 7^\circ \), \( a \), and \( 2a - 2^\circ \)? Wait, no, wait, the diagram shows a triangle? Wait, no, the exterior angles? Wait, maybe it's a triangle, and the exterior angles? Wait, no, the sum of exterior angles of a triangle? Wait, no, the sum of exterior angles of any polygon is \( 360^\circ \), but for a triangle, the sum of exterior angles (one at each vertex) is \( 360^\circ \). Wait, but in this case, maybe the three angles given are the exterior angles? Wait, no, looking at the diagram, maybe it's a triangle with three exterior angles? Wait, no, the angles are \( 2a + 7^\circ \), \( a \), and \( 2a - 2^\circ \), and maybe another angle? Wait, no, the diagram shows three angles? Wait, no, the problem says "the diagram shows a convex polygon" – maybe it's a triangle? Wait, no, the sum of exterior angles of any polygon is \( 360^\circ \), but if it's a triangle, the sum of exterior angles is \( 360^\circ \). Wait, but in the diagram, maybe there are three angles? Wait, no, the angles given are \( 2a + 7^\circ \), \( a \), and \( 2a - 2^\circ \), and maybe a straight line? Wait, no, perhaps the three angles are the exterior angles, and we need to sum them to \( 360^\circ \)? Wait, no, that can't be, because \( 2a + 7 + a + 2a - 2 = 5a + 5 \), and if that's \( 360 \), then \( 5a = 355 \), \( a = 71 \), but that seems high. Wait, maybe it's a triangle, and the sum of exterior angles? No, wait, maybe the diagram is a triangle with three exterior angles, but that's not right. Wait, no, maybe the angles are the exterior angles of a triangle, but the sum of exterior angles of a triangle is \( 360^\circ \)? Wait, no, the sum of interior angles of a triangle is \( 180^\circ \), and each exterior angle is supplementary to the interior angle. Wait, maybe the diagram is a triangle with three angles, but the angles given are the exterior angles? Wait, no, let's re-examine. Wait, the problem says "the diagram shows a convex polygon" – maybe it's a triangle, and the three angles are the exterior angles? Wait, no, the sum of exterior angles of any polygon is \( 360^\circ \), so if there are three exterior angles, their sum is \( 360^\circ \). Wait, but the angles are \( 2a + 7^\circ \), \( a \), and \( 2a - 2^\circ \). Wait, but that would be three angles, sum to \( 360 \). So:

\( (2a + 7) + a + (2a - 2) = 360 \)

Wait, but that would be \( 5a + 5 = 360 \), \( 5a = 355 \), \( a = 71 \). But that seems too big. Wait, maybe it's a triangle, and the sum of the exterior angles is \( 360 \), but maybe the diagram has three angles, but I'm missing something. Wait, no, maybe the diagram is a triangle with three angles, but the angles given are the exterior angles? Wait, no, perhaps the problem is that the three angles are the exterior angles, and we need to sum them to \( 360 \). Wait, but let's check again.

Wait, maybe the diagram is a triangle, and the three angles are the exterior angles, so:

\( (2a + 7) + a + (2a - 2) = 360 \)

\( 5a + 5 = 360 \)

\( 5a = 355 \)

\( a = 71 \). But that seems high. Wait, maybe the diagram is a triangle, and the sum of the exterior angles is \( 360 \), but maybe I made a mistake. Wait, no, the sum of exterior angles of any polygon is \( 360^\circ \), regardless of the number of sides. So if it's a triangle, three exterior angles, sum to \( 360 \). So that's the equation.

Wait, but let's check the calculation:

\( 2a + 7 + a + 2a - 2 = 5a + 5 \)

Set equal to \( 360 \):

\( 5a + 5 = 360 \)

Subtract 5:

\( 5a = 355 \)

Divide by 5:

\( a = 71 \)

But let's verify. If \( a = 71 \), then the angles are:

\( 2(71) + 7 = 142 + 7 = 149^\circ \)

\( a = 71^\circ \)

\( 2(71) - 2 = 142 - 2 = 140^\circ \)

Sum: \( 149 + 71 + 140 = 360 \). Yes, that works. So \( a = 71 \).

Wait, but maybe the diagram is a triangle, and the three angles are the exterior angles. So the sum is \( 360^\circ \). So the equation is correct.

Step1: Set up the equation for the sum of exterior angles

The sum of the exterior angles of any convex polygon is \( 360^\circ \). The given angles are \( 2a + 7^\circ \), \( a \), and \( 2a - 2^\circ \). So we set up the equation:

\( (2a + 7) + a + (2a - 2) = 360 \)

Step2: Simplify the left side

Combine like terms:

\( 2a + a + 2a + 7 - 2 = 5a + 5 \)

So the equation becomes:

\( 5a + 5 = 360 \)

Step3: Solve for \( a \)

Subtract 5 from both sides:

\( 5a = 360 - 5 \)

\( 5a = 355 \)

Divide both sides by 5:

\( a = \frac{355}{5} \)

\( a = 71 \)

Answer:

\( a = \boxed{71} \)