Question

for the triangle at right, determine the measures of ∠a and ∠b. once you have solved for the measure of the first acute angle (either ∠a or ∠b), what knowledge about the angles in triangles could help you solve for the second acute angle measure? examine the triangle below. note: the diagrams are not drawn to scale. (diagram shows a right triangle with right angle at c, side ac labeled 14, hypotenuse ab labeled 15)

Explanation:

Step1: Identify triangle type

It's a right - triangle with \( \angle C = 90^{\circ} \), \( AC = 14 \), \( AB=15 \). We can use trigonometric ratios or the Pythagorean theorem. First, find \( BC \) using Pythagorean theorem: \( BC=\sqrt{AB^{2}-AC^{2}}=\sqrt{15^{2}-14^{2}}=\sqrt{225 - 196}=\sqrt{29}\approx5.39 \)

Step2: Find \( \angle A \)

Using sine function: \( \sin A=\frac{BC}{AB}=\frac{\sqrt{29}}{15}\approx\frac{5.39}{15}\approx0.359 \)
\( \angle A=\arcsin(0.359)\approx21.0^{\circ} \)

Step3: Find \( \angle B \)

In a right - triangle, \( \angle A+\angle B = 90^{\circ} \), so \( \angle B=90^{\circ}-\angle A\approx90 - 21.0 = 69.0^{\circ} \)

Step4: Solve for the second acute angle

Once we have one acute angle (say \( \angle A \)), we know that in a right - triangle, the sum of the two acute angles is \( 90^{\circ} \) (angle sum property of triangles: the sum of interior angles of a triangle is \( 180^{\circ} \), and since one angle is \( 90^{\circ} \), the other two sum to \( 90^{\circ} \)). So if we know \( \angle A \), we can find \( \angle B = 90^{\circ}-\angle A \) and vice - versa.

Answer:

\( \angle A\approx21.0^{\circ} \), \( \angle B\approx69.0^{\circ} \); The angle sum property of triangles (sum of acute angles in right - triangle is \( 90^{\circ} \)) helps find the second acute angle.