Given a right triangle with angles $A$, $B$, and $C$, where $C$ is the right angle, if $m\angle A=m\angle B$ and $m\angle A+m\angle C=m\angle D$, what is the measure of angle $D$?

• A. $180°$
• B. $90°$
• C. $60°$
• D. $135°$

Answer: (D)

Verified step by step guidance

1. Identify the given information: triangle ABC is a right triangle with angle C as the right angle, so \(m\angle C = 90^\circ\).
2. Since \(m\angle A = m\angle B\) and the sum of angles in any triangle is \(180^\circ\), write the equation: \(m\angle A + m\angle B + m\angle C = 180^\circ\).
3. Substitute \(m\angle A = m\angle B\) and \(m\angle C = 90^\circ\) into the equation to get: \(m\angle A + m\angle A + 90^\circ = 180^\circ\).
4. Solve for \(m\angle A\): \(2 m\angle A = 180^\circ - 90^\circ\), so \(m\angle A = 45^\circ\). Since \(m\angle A = m\angle B\), \(m\angle B = 45^\circ\) as well.
5. Use the given relation \(m\angle A + m\angle C = m\angle D\) to find \(m\angle D\): substitute \(m\angle A = 45^\circ\) and \(m\angle C = 90^\circ\) to get \(m\angle D = 45^\circ + 90^\circ\).