Question

multiple choice 4 points
the measure of one acute angle of an obtuse triangle is 20° more than the measure
of the other acute angle. what is a possible measure of the smaller acute angle?
30° 45° 40° 35°
multiple choice 4 points
two angles of a triangle add up to 65°. what is the measure of the third angle?
115° 25° 295° 55°

Explanation:

Step1: Solve the third - angle problem

Let the two angles be \(x\) and \(y\), and we know \(x + y=65^{\circ}\). The sum of angles in a triangle is \(180^{\circ}\). So the third - angle \(z = 180-(x + y)\). Substitute \(x + y = 65^{\circ}\) into the formula, we get \(z=180 - 65=115^{\circ}\).

Step2: Solve the acute - angle problem

Let the smaller acute angle be \(a\), and the larger acute angle be \(b\). We know \(b=a + 20^{\circ}\), and \(a + b<90^{\circ}\) (because the triangle is obtuse, the sum of the two acute angles is less than \(90^{\circ}\)). Substitute \(b=a + 20^{\circ}\) into \(a + b<90^{\circ}\), we have \(a+(a + 20)<90\). Simplify the inequality: \(2a+20<90\), \(2a<70\), \(a < 35^{\circ}\). Among the options \(30^{\circ}\) satisfies this condition.

Answer:

For the third - angle problem: \(115^{\circ}\). For the acute - angle problem: \(30^{\circ}\)