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.

m∠ A+m∠ B+m∠ C=180

SubEqn

LHS-(m∠ A+m∠ B)=RHS-(m∠ A+m∠ B)

m∠ C=180- (m∠ A+m∠ B)

We know that m∠ A+m∠ B>90. Let's see what this implies about the measure of ∠ C.

m∠ A+m∠ B>90

MultNegIneq

Multiply by -1 and flip inequality sign

-(m∠ A+m∠ B)<-90

AddIneq

LHS+180

180-(m∠ A+m∠ B)<180-90

SubTerms

Subtract terms

180-(m∠ A+m∠ B)<90

Substitute

180-(m∠ A+m∠ B)= m∠ C

m∠ C<90

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.