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 correct. the remaining angle of the triangle measures less than 90 degrees. she is incorrect. the angles measure greater than 90 degrees so the triangle is obtuse. she is incorrect. the third angle in a triangle with two congruent acute angles is a right angle.

Explanation:

Let the two congruent acute angles be \(x\) each. So their sum is \(2x>90^{\circ}\), which means \(x > 45^{\circ}\).
The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(y\). Then \(y=180^{\circ}-2x\).
Since \(2x>90^{\circ}\), then \(y = 180^{\circ}-2x<90^{\circ}\).
An acute - angled triangle is a triangle in which all angles are less than \(90^{\circ}\). Here, all three angles (\(x,x,y\)) are less than \(90^{\circ}\) (because \(x\) is acute (\(x < 90^{\circ}\)) and \(y=180 - 2x<90^{\circ}\)).

Answer:

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