what are the angle measures in triangle abc?\n$m\\angle a = 90^{\\circ}, m\\angle b = 30^{\\circ}, m\\angle…

what are the angle measures in triangle abc?\n$m\\angle a = 90^{\\circ}, m\\angle b = 30^{\\circ}, m\\angle c = 60^{\\circ}$\n$m\\angle a = 60^{\\circ}, m\\angle b = 90^{\\circ}, m\\angle c = 30^{\\circ}$\n$m\\angle a = 90^{\\circ}, m\\angle b = 60^{\\circ}, m\\angle c = 30^{\\circ}$\n$m\\angle a = 60^{\\circ}, m\\angle b = 30^{\\circ}, m\\angle c = 90^{\\circ}$

what are the angle measures in triangle abc?\n$m\\angle a = 90^{\\circ}, m\\angle b = 30^{\\circ}, m\\angle c = 60^{\\circ}$\n$m\\angle a = 60^{\\circ}, m\\angle b = 90^{\\circ}, m\\angle c = 30^{\\circ}$\n$m\\angle a = 90^{\\circ}, m\\angle b = 60^{\\circ}, m\\angle c = 30^{\\circ}$\n$m\\angle a = 60^{\\circ}, m\\angle b = 30^{\\circ}, m\\angle c = 90^{\\circ}$

Answer

Explanation:

Step1: Check the Pythagorean theorem

In a right - triangle, (a^{2}+b^{2}=c^{2}). Let (a = 6), (b=6\sqrt{3}), (c = 12). (a^{2}=6^{2}=36), (b^{2}=(6\sqrt{3})^{2}=36\times3 = 108), (c^{2}=12^{2}=144). Since (36 + 108=144), by the Pythagorean theorem, (\angle A=90^{\circ}).

Step2: Use the sine function

The sine function is defined as (\sin C=\frac{\text{opposite}}{\text{hypotenuse}}). In (\triangle ABC), for (\angle C), the opposite side to (\angle C) is (AB = 6) and the hypotenuse (BC = 12). So (\sin C=\frac{AB}{BC}=\frac{6}{12}=\frac{1}{2}). Since (\sin30^{\circ}=\frac{1}{2}), then (m\angle C = 30^{\circ}).

Step3: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is (180^{\circ}). Let (m\angle A = 90^{\circ}), (m\angle C=30^{\circ}). Using the formula (m\angle A+m\angle B+m\angle C = 180^{\circ}), we substitute the known values: (90^{\circ}+m\angle B + 30^{\circ}=180^{\circ}). Solving for (m\angle B): (m\angle B=180^{\circ}-(90^{\circ}+30^{\circ})=60^{\circ}).

Answer:

(m\angle A = 90^{\circ},m\angle B = 60^{\circ},m\angle C = 30^{\circ}) (the third option)