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.

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\). Let the two acute angles be \(A\) and \(B\) (so \(A < 90^\circ\), \(B < 90^\circ\)) and the third angle be \(C\). Then \(A + B + C = 180^\circ\), so \(C = 180^\circ - (A + B)\).

Step2: Analyze Given Condition (\(A + B > 90^\circ\))

If \(A + B > 90^\circ\), substitute into \(C\)'s formula: \(C = 180^\circ - (A + B) < 180^\circ - 90^\circ = 90^\circ\). So \(C < 90^\circ\).

Step3: Determine Triangle Type

Since all three angles (\(A < 90^\circ\), \(B < 90^\circ\), \(C < 90^\circ\)) are acute (less than \(90^\circ\)), the triangle is acute.

Answer:

The statement is true. Because the sum of angles in a triangle is \(180^\circ\). If the sum of two acute angles (\(A + B\)) is greater than \(90^\circ\), the third angle \(C = 180^\circ - (A + B)\) is less than \(90^\circ\). Thus, all three angles are acute, so the triangle is acute.