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:

1. Recall the angle - sum property of a triangle: The sum of the interior angles of a triangle is \(180^{\circ}\).
2. Let the two congruent acute angles be \(x\) each. We know that \(x + x>90^{\circ}\), so \(2x>90^{\circ}\), or \(x > 45^{\circ}\).
3. Let the third angle be \(y\). Then, by the angle - sum property of a triangle, \(x + x+y=180^{\circ}\), so \(y = 180^{\circ}-2x\).
4. Since \(2x>90^{\circ}\), then \(y=180^{\circ}-2x < 180^{\circ}-90^{\circ}=90^{\circ}\). Also, since \(x\) is acute (\(x < 90^{\circ}\)) and \(y<90^{\circ}\), all three angles of the triangle are acute. So Anna is correct because the remaining angle \(y\) measures less than \(90^{\circ}\).

• Analyze the first option: A triangle with at least one acute angle is not necessarily an acute triangle. For example, a right - angled triangle has two acute angles and one right angle, and an obtuse - angled triangle has two acute angles and one obtuse angle. So this option is incorrect.
• Analyze the third option: The sum of the two acute angles is greater than \(90^{\circ}\), so the third angle \(y = 180-(x + x)<90^{\circ}\), so the triangle is not obtuse. This option is incorrect.
• Analyze the fourth option: The third angle \(y=180 - 2x\). Since \(2x>90^{\circ}\), \(y < 90^{\circ}\), not a right angle. This option is incorrect.

Answer:

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