Question

determine which of the following best describes the truth value and a counterexample, if applicable, of the statement \if two angles in a triangle are acute, then it is an acute triangle.\ two of these are correct. this statement is false. a counterexample would be a triangle with angles that are 42, 10, and 128 degrees. this statement is true. this statement is false. a counterexample would be a triangle with angles that are 45, 45, and 90 degrees.

Explanation:

Step1: Recall triangle - angle definitions

An acute - angled triangle has all angles less than 90 degrees. An acute angle is an angle with measure between 0 and 90 degrees.

Step2: Analyze the given statement

The statement "If two angles in a triangle are acute, then it is an acute triangle" is being evaluated.

Step3: Check the counter - examples

• For a triangle with angles 42, 10, and 128 degrees: 42 and 10 are acute, but 128 is obtuse. However, this triangle has an error as the sum of these angles \(42 + 10+128=180\) is incorrect for a triangle (the sum of interior angles of a triangle must be 180 degrees, and \(42 + 10+128 = 180\) is wrong arithmetic).
• For a triangle with angles 45, 45, and 90 degrees: The angles 45 and 45 are acute, but the angle 90 is a right - angle. So this is a right - angled triangle, not an acute - angled triangle, and it is a valid counter - example as \(45 + 45+90=180\) degrees. Since we have a counter - example, the statement is false.

Answer:

This statement is false. A counterexample would be a triangle with angles that are 45, 45, and 90 degrees.