Given that $m\angle A+m\angle B=m\angle B+m\angle C$ in triangle $\mathrm{ABC}$, which of the following must be true?

• A. $m\angle B=m\angle C$
• B. $m\angle A+m\angle B+m\angle C=90°$
• C. $m\angle A=m\angle B$
• D. $m\angle C=m\angle A$

Answer: (D)

Verified step by step guidance

1. Start with the given equation: \(m\angle A + m\angle B = m\angle B + m\angle C\).
2. Subtract \(m\angle B\) from both sides to isolate the angles: \(m\angle A = m\angle C\).
3. Recall that in any triangle \(ABC\), the sum of the interior angles is \$180^\circ\(, i.e., \)m\angle A + m\angle B + m\angle C = 180^\circ$.
4. Since \(m\angle A = m\angle C\), you can substitute \(m\angle C\) with \(m\angle A\) in the angle sum equation to find relationships between the angles if needed.
5. Conclude that the given condition implies \(m\angle A = m\angle C\), which means angles \(A\) and \(C\) are equal.