Question

analyzing triangle classifications
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.
she is incorrect. the third angle in a triangle with two congruent acute angles is a right angle.

Explanation:

Step1: Recall triangle - angle sum property

The sum of the interior angles of a triangle is 180°. Let the two congruent acute angles each have measure \(x\) and the third angle have measure \(y\). So \(2x + y=180^{\circ}\), and we know \(2x>90^{\circ}\).

Step2: Solve for the third - angle \(y\)

We can rewrite the equation \(2x + y = 180^{\circ}\) as \(y=180^{\circ}-2x\). Since \(2x>90^{\circ}\), then \(y = 180 - 2x<90^{\circ}\). An acute - angled triangle has all angles less than 90°. Since all three angles (\(x,x,y\)) are less than 90°, Anna is correct.

Answer:

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