Question

In triangle MNP, angle M = 30 degrees, angle N = 20 degrees, and MN = 10 units. What is the approximate length of PN?

Answer 1

M=30°; N=20°; MN=10 units; PN=?
In this case, we have one side MN=10 units, and the two adjacent angles, angle M=30° and angle N=20°. This is an ASA triangle (Angle Side Angle), and we can solve it using the Law of Sines:
PN / sin M=MN / sin P 
PN / sin 30°=10 / sin P
angle P=?

angle M + angle N + angle P=180°
30°+20°+angle P=180°
50°+angle P=180°
Solving for angle P:
50°+angle P -50°=180°-50°
angle P=130°

Replacing in the Law of sines:
PN / sin 30°=10 / sin P
PN / sin 30°=10 / sin 130°
Solving for PN. Multiplying both sides of the equation by sin 30°:
sin 30°(PN/sin 30°)=sin 30°(10/sin 130°)
PN=10 sin 30°/sin 130°
PN=10(0.5)/0.766044443
PN=6.527036448
Rounded to one decimal place:
PN=6.5

Answer: Second option 6.5 units