Answer:
Explanation:
In a triangle, the sum of the measures of the three angles is always 180°. So, if m∠B = 111°, then m∠A + m∠C = 180° - 111° = 69°.
We are also given that a = 12 and c = 6. These two side lengths can be used to find the value of the third side length (b) using the Law of Cosines.
The Law of Cosines states that:
b² = a² + c² - 2ac cos(B)
Plugging in the known values, we get:
b² = 12² + 6² - 2 * 12 * 6 * cos(111°)
b² = 144 + 36 - 144 * (-0.396)
b² = 216 + 56.64
b = 15.2
Now that we know the value of b, we can find the measures of m∠A and m∠C using the following equations:
m∠A = arccos( (b² + c² - a²) / (2bc) )
m∠C = 180° - m∠A - m∠B
Plugging in the known values, we get:
m∠A = arccos( (15.2² + 6² - 12²) / (2 * 15.2 * 6) )
m∠A = arccos( 0.796 )
m∠A = 44.8°
m∠C = 180° - 44.8° - 111° = 24.2°
Therefore, the measures of the remaining side and angles are m∠A = 44.8°, m∠C = 24.2°, and b = 15.2.