Question

estimate the measures of the angles. round to the nearest tenth.
type your answers in the boxes.
m∠c =

m∠b =

(image of triangle abc with right angle at a, ac = 6.2, ab = 3.7, bc = 7.2)

Explanation:

Step1: Identify triangle type and trigonometric ratios

Triangle \(ABC\) is right - angled at \(A\) (\(\angle A = 90^{\circ}\)). We can use trigonometric ratios (tangent, sine, cosine) to find the other angles. Let's first find \(\angle B\) and \(\angle C\).

For \(\angle B\): In right - triangle \(ABC\), \(\tan B=\frac{AC}{AB}\). Given \(AC = 6.2\) and \(AB=3.7\). So \(\tan B=\frac{6.2}{3.7}\approx1.6757\). Then \(m\angle B=\arctan(1.6757)\). Using a calculator, \(\arctan(1.6757)\approx59.2^{\circ}\) (rounded to the nearest tenth).

For \(\angle C\): We know that in a right - triangle, \(\angle A+\angle B+\angle C = 180^{\circ}\), and \(\angle A = 90^{\circ}\). So \(\angle C=90^{\circ}-\angle B\). Since \(\angle B\approx59.2^{\circ}\), then \(\angle C = 90 - 59.2=30.8^{\circ}\). We can also use \(\tan C=\frac{AB}{AC}=\frac{3.7}{6.2}\approx0.5968\), and \(\arctan(0.5968)\approx30.8^{\circ}\) (rounded to the nearest tenth).

Step2: Verify the calculations

• For \(\angle B\): \(\tan(59.2^{\circ})\approx1.67\), and \(\frac{6.2}{3.7}\approx1.676\), which is close.
• For \(\angle C\): \(\tan(30.8^{\circ})\approx0.597\), and \(\frac{3.7}{6.2}\approx0.597\), which is close. Also, since \(\angle A = 90^{\circ}\), \(\angle B+\angle C=59.2 + 30.8 = 90^{\circ}\), which satisfies the angle - sum property of a triangle.

Answer:

\(m\angle C\approx30.8^{\circ}\), \(m\angle B\approx59.2^{\circ}\)