Question

which of the following belongs in the blank in the geometric proof shown below? statement reason $\triangle abc$ is a right triangle with $angle c$ as the right angle. given $mangle c = 90^{circ}$ definition of right angle ____________ triangle sum theorem $mangle a + mangle b = 90^{circ}$ subtracting property of equality $angle a$ and $angle b$ are complementary angles definition of complementary angles a $angle b = 45^{circ}$ b $0^{circ}

Explanation:

Step1: Recall Triangle Sum Theorem

The sum of the interior - angles of a triangle is 180°. In \(\triangle ABC\), the angles are \(\angle A\), \(\angle B\), and \(\angle C\).
So, \(m\angle A + m\angle B+m\angle C=180^{\circ}\).

Step2: Analyze the proof steps

The step before the blank uses the Triangle Sum Theorem. The theorem states that for any triangle, the sum of the measures of its interior angles is 180°. In the context of \(\triangle ABC\) with \(\angle C = 90^{\circ}\), the statement based on the Triangle Sum Theorem is \(m\angle A + m\angle B+m\angle C = 180^{\circ}\).

Answer:

D. \(m\angle A + m\angle B + m\angle C=180^{\circ}\)