Question

How many distinct triangles can be formed for which m∠E = 64°, g = 9, and e = 10?

triangle(s)

How many distinct triangles can be formed for which m∠J = 129°, k = 8, and j = 3?
triangle(s)

Answer 1

1

0

Explanation:

Answer 2

• 1
• 0

Explanation:

1.

For two side measures and one angle measure to form one distinct triangle, the given angle must be between the given sides, or opposite the longest side.

When the given measures are ...

• E = 64°
• g = 9
• e = 10

The given angle is opposite the longest side, so one distinct triangle can be formed.

2.

No triangle can be formed if the given angle is opposite the shorter given side, and either of ...

• the angle is right or obtuse
• the ratio of shorter to longer sides is less than the sine of the given angle.

__

Additional comment

As we can see from the above, the only ambiguous cases arise when the given angle is opposite the shorter given side. When that happens, the ratio of the shorter to longer given side relative to the sine of the given angle determines the nature of the solution:

• two distinct triangles if the ratio is greater than the sine
• one right triangle if the ratio is equal to the sine.