Question

in triangle abc, the measure of angle a is x°, the measure of angle b is 20°, and ab = 37. which additional piece of information is sufficient to prove that triangle abc is a right triangle?
a x = 70
b x = 160
c ac = 37
d ac = 53

Explanation:

Step1: Recall the sum of angles in a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\).

Step2: Analyze option A

If \(x = 70\), then the third - angle \(C=180-(70 + 20)=90^{\circ}\).

Step3: Analyze option B

If \(x = 160\), then \(160+20=180\), which is not possible for a triangle (a triangle has three angles).

Step4: Analyze option C

If \(AC = 37\) (i.e., \(AB = AC = 37\)), then the triangle is isosceles with \(\angle B=\angle C = 20^{\circ}\), and \(\angle A=140^{\circ}\), not a right - triangle.

Step5: Analyze option D

Just knowing \(AC = 53\) and \(AB = 37\) and \(\angle B=20^{\circ}\) is not enough to use the Pythagorean theorem (we don't know if the sides satisfy \(a^{2}+b^{2}=c^{2}\)) without more angle information.

Answer:

A. \(x = 70\)