Question

construct arguments determine whether the following statement is true or false. if false, give a counterexample. if true, give an argument to support your conclusion.
if the sum of two acute angles of a triangle is greater than 90°,
then the triangle is acute.
because the sum of the two acute angles is greater than 90°, the measure of the third angle is a number greater than select choice subtracted from select choice, which must be select choice. therefore, the triangle has select choice acute angles and select choice acute select choice acute

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\). Let the two acute angles be \(A\) and \(B\), and the third angle be \(C\). So \(A + B + C=180^\circ\).

Step2: Analyze Given Condition

Given \(A + B>90^\circ\). Substitute into the angle - sum formula: \(C = 180^\circ-(A + B)\).

Step3: Determine Measure of Third Angle

Since \(A + B>90^\circ\), then \(C=180^\circ-(A + B)<180^\circ - 90^\circ=90^\circ\). Also, \(A\) and \(B\) are acute (\(A<90^\circ\), \(B<90^\circ\)) and \(A + B>90^\circ\), so all three angles (\(A\), \(B\), \(C\)) are acute (less than \(90^\circ\)). So the statement is true. The blanks should be filled as follows: "Because the sum of the two acute angles is greater than \(90^\circ\), the measure of the third angle is a number greater than \(0^\circ\) (wait, no, from \(C = 180-(A + B)\) and \(A + B>90\), so \(C<90\)) and less than \(90^\circ\) (since \(A + B>90\), \(C = 180-(A + B)<90\)). Wait, the first blank: "subtracted from \(180^\circ\)" (because \(C=180-(A + B)\)), the second blank: "the measure of the third angle is a number greater than \(0^\circ\)?" No, let's re - express. The sum of two acute angles (\(A + B\)) is subtracted from \(180^\circ\) to get the third angle \(C\). Since \(A + B>90^\circ\), then \(C = 180-(A + B)<90^\circ\). And since \(A\) and \(B\) are acute (\(A<90\), \(B<90\)) and \(A + B>90\), all three angles are acute. So the argument: In a triangle, angle sum is \(180^\circ\). If two acute angles sum to more than \(90^\circ\), the third angle is less than \(90^\circ\) (as \(180 - (\text{sum of two acutes})<90\)). Since the two given angles are acute (less than \(90\)) and their sum is more than \(90\), all three angles are less than \(90\), so the triangle is acute.

Answer:

The statement "If the sum of two acute angles of a triangle is greater than \(90^\circ\), then the triangle is acute" is True. The blanks (from the context of the fill - in - the - blanks) should be filled such that we use the triangle angle - sum property to show the third angle is acute and thus the triangle is acute.