A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than...

Understanding the Question

We need to determine whether all three angles in a triangle with sides a, b, and c are less than 90 degrees. In other words: Is this an acute triangle?

Here's the key geometric insight: A triangle is acute (all angles < 90°) if and only if the square of the longest side is less than the sum of squares of the other two sides. If c is the longest side, we need: \(\mathrm{c^2 < a^2 + b^2}\)

What makes this question elegant is that we're not asked to find the actual angle measures—we just need to determine if we can definitively answer yes or no to whether they're all less than 90°.

Analyzing Statement 1

Statement 1 tells us that semicircles drawn on each side of the triangle have areas of 3, 4, and 6 cm² respectively.

This connects to a beautiful geometric theorem: When you draw semicircles on the sides of a triangle, the relationship between their areas reveals the triangle's angle types.

Here's the classical result:

• If the sum of the two smaller semicircle areas equals the largest area → Right triangle
• If the sum exceeds the largest area → Acute triangle
• If the sum is less than the largest area → Obtuse triangle

Let's apply this:

• The three semicircle areas are 3, 4, and 6 cm²
• Sum of two smaller areas: \(\mathrm{3 + 4 = 7}\) cm²
• Largest area: 6 cm²
• Since \(\mathrm{7 > 6}\), the triangle is acute

Therefore, we can definitively say YES—all angles are less than 90°.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(\mathrm{c < a + b < c + 2}\)

This gives us two pieces of information:

• \(\mathrm{c < a + b}\) (confirms we have a valid triangle—the triangle inequality)
• \(\mathrm{a + b < c + 2}\) (the sum of two sides is less than 2 units more than the third side)

But does this narrow constraint tell us whether \(\mathrm{c^2 < a^2 + b^2}\)?

Let's test with specific examples:

Example 1 (Obtuse Triangle): Let \(\mathrm{c = 5, a = 3, b = 3}\)

• Check constraint: \(\mathrm{c < a + b < c + 2 \rightarrow 5 < 6 < 7}\) ✓
• Check angle type: \(\mathrm{c^2 = 25, a^2 + b^2 = 18}\)
• Since \(\mathrm{25 > 18}\), this triangle has an obtuse angle

Example 2 (Acute Triangle): Let \(\mathrm{c = 4, a = 3, b = 3}\)

• Check constraint: \(\mathrm{c < a + b < c + 2 \rightarrow 4 < 6 < 6}\) ✓
• Check angle type: \(\mathrm{c^2 = 16, a^2 + b^2 = 18}\)
• Since \(\mathrm{16 < 18}\), this triangle is acute

We found both an obtuse triangle and an acute triangle that satisfy Statement 2's constraint. Therefore, we cannot determine whether all angles are less than 90°.

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Statement 1 alone is sufficient because the semicircle theorem gives us a definitive YES answer—the triangle is acute. Statement 2 alone is not sufficient because triangles with different angle types can satisfy the given constraint.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."