Problem

Compare the triangle areas when the \(\text{same}~\text{two}~\text{sides}\), \(a\) and \(b\), are fixed and the included angles are \(30^{\circ}\) and \(150^{\circ}\).

Two triangles with the same sides a and b and included angles 30 and 150 degrees, each with a perpendicular altitude.

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What this problem is really about

With two side lengths fixed, the constant factor one-half times their product is the same for every configuration. Compare areas by comparing only the sine of each included angle; supplementary angles have equal sine values and therefore equal height magnitudes, even when one altitude falls outside the triangle. Area reaches its maximum when the sine factor is one.

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The triangle with a 150 degree angle has the larger area because a larger included angle always makes a larger triangle when the two sides are fixed.

The triangle with a 30 degree angle has the larger area because the 150 degree case is obtuse, so its altitude falls outside the triangle and the area becomes smaller.

The areas are equal because with fixed sides the area is one-half ab sin(C), and sin 30 degrees equals sin 150 degrees.

The triangle with a 150 degree angle has the larger area because sin 150 degrees is greater than sin 30 degrees when the angle is greater.

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Compare the triangle areas when the \(\text{same}~\text{two}~\text{sides}\), \(a\) and \(b\), are fixed and the included angles are \(30^{\circ}\) and \(150^{\circ}\).

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