Question

determine whether each statement is true.
select true or false for each statement.
a triangle can have more than two acute angles.
the sum of the measures of the interior angles of a triangle is always 360°.
if a triangle has an obtuse angle, the sum of the measures of the acute angles is greater than 90°.
the sum of the measures of the exterior angles at each vertex of a triangle is 360°.

Explanation:

1. Statement: A triangle can have more than two acute angles.

A triangle's interior angles sum to \(180^\circ\). An acute angle is less than \(90^\circ\). If a triangle has three acute angles (e.g., an equilateral triangle with all angles \(60^\circ\)), the sum \(60 + 60 + 60 = 180^\circ\), which is valid. So a triangle can have three (more than two) acute angles. Thus, the statement is True.

By the triangle angle - sum theorem, the sum of the interior angles of a triangle is \(180^\circ\), not \(360^\circ\) (the sum of exterior angles of a polygon with \(n\) sides is \(360^\circ\), for \(n = 3\), exterior angle sum is \(360^\circ\), but interior angle sum of triangle is \(180^\circ\)). So the statement is False.

Let the obtuse angle be \(O\) (\(O>90^\circ\)) and the two acute angles be \(A\) and \(B\) (\(A < 90^\circ\), \(B < 90^\circ\)). Since \(A + B+O=180^\circ\) and \(O > 90^\circ\), then \(A + B=180^\circ - O\). As \(O>90^\circ\), \(180^\circ - O<90^\circ\). So the sum of acute angles is less than \(90^\circ\), not greater. The statement is False.

Answer:

True

2. Statement: The sum of the measures of the interior angles of a triangle is always \(360^\circ\).