Question

triangle def contains two congruent acute angles. the sum of the measures of the two congruent acute angles is greater than 90 degrees. anna concludes that the triangle must be an acute triangle. which best describes her conclusion? she is correct. a triangle having at least one acute angle is an acute triangle. she is incorrect. the angles measure greater than 90 degrees so the triangle is obtuse. she is correct. the remaining angle of the triangle measures less than 90 degrees. she is incorrect. the third angle in a triangle with two congruent acute angles is a right angle.

Explanation:

Step1: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let the two congruent acute angles be \(x\) each. So the sum of these two angles is \(2x\), and the third angle is \(y\). Then \(2x + y=180^{\circ}\), or \(y = 180^{\circ}-2x\)

Step2: Analyze the given condition

We know that \(2x>90^{\circ}\). Then \(y=180^{\circ}-2x\). Since \(2x > 90^{\circ}\), we can find the range of \(y\) by subtracting \(2x\) from \(180^{\circ}\).

If \(2x>90^{\circ}\), then \(y=180 - 2x<180 - 90\). So \(y < 90^{\circ}\)

Step3: Analyze each option

• Option 1: A triangle having at least one acute angle is not an acute - triangle. An acute - triangle has all angles less than \(90^{\circ}\). So this option is wrong.
• Option 2: Since \(y=180 - 2x\) and \(2x>90\), \(y<90\). So the triangle is not obtuse. This option is wrong.
• Option 3: As shown above, if \(2x>90\), then \(y = 180 - 2x<90\). All angles (\(x,x,y\)) are less than \(90^{\circ}\) (because \(x\) is acute, \(x<90\) and \(y<90\)). So this option is correct.
• Option 4: If \(2x>90\), then \(y=180 - 2x<90\). The third angle is not a right - angle. So this option is wrong.

Answer:

She is correct. The remaining angle of the triangle measures less than 90 degrees.