Question

determining the number of possible triangles

how many distinct triangles can be formed for which (m\angle e = 64^{\circ}), (g = 9), and (e = 10)?
triangle(s)

how many distinct triangles can be formed for which (m\angle j = 129^{\circ}), (k = 8), and (j = 3)?
triangle(s)

Explanation:

🆕 New Concept Discovered: The Ambiguous Case (SSA) of the Law of Sines
When we know two sides and a non-included angle, there might be 0, 1, or 2 possible triangles.

Step 1: Analyze the first scenario

We are given:

• Angle \( E = 64^\circ \) (an acute angle)
• Side adjacent to the angle, \( g = 9 \)
• Side opposite to the angle, \( e = 10 \)

Since the opposite side \( e \) is greater than the adjacent side \( g \) (\( 10 > 9 \)), the opposite side is long enough to reach the base in exactly one way.

Therefore, there is exactly 1 distinct triangle.

Step 2: Analyze the second scenario

We are given:

• Angle \( J = 129^\circ \) (an obtuse angle)
• Side adjacent to the angle, \( k = 8 \)
• Side opposite to the angle, \( j = 3 \)

For an obtuse angle, the side opposite the obtuse angle must be the longest side of the triangle. This means we must have \( j > k \).

Since \( j = 3 \) and \( k = 8 \), we have \( 3 < 8 \), which is impossible for a valid triangle.

Therefore, there are 0 distinct triangles.

Answer:

First box: 1
Second box: 0