Exercise 40 Page 341

We are asked to determine whether the following statement is true or false.

If the sum of the measure of two acute angles of a triangle is greater than 90, then the triangle is acute.

Let's see what we can conclude about the third angle in a triangle, where the hypothesis is true. Let the acute angles of triangle △ ABC be ∠ A and ∠ B. The Triangle Angle-Sum Theorem gives a connection between the interior angle measures of a triangle.

Triangle Angle-Sum Theorem

The sum of the measures of the interior angles of a triangle is 180.

Let's rearrange this relationship to express one of the angle measures in terms of the other two. We know that m∠ A+m∠ B>90. Let's see what this implies about the measure of ∠ C. Thus, if the sum of the measures of two acute angles in a triangle is greater than 90, then the third angle is acute. This means that the triangle has three acute angles, so it is an acute triangle. The statement is true.