Question

you are shown $\angle cde$ below whose measure is $114^\circ$. draw an angle bisector of $\angle cde$ at $d$.
click and drag to draw a ray from the yellow dot.
to redraw, click and drag from the yellow dot again.
$\text{m}\angle cde = 114^\circ$ $\text{m}\angle cdf = 57^\circ$ $\text{m}\angle edf = 57^\circ$
which of the following statements is true?
answer
$\bigcirc$ $\frac{1}{2}(\text{m}\angle cdf) = \text{m}\angle edf$ $\bigcirc$ $\text{m}\angle cde = \text{m}\angle edf$
$\bigcirc$ $\text{m}\angle cde = 2(\text{m}\angle edf)$ $\bigcirc$ $\text{m}\angle cdf = \frac{1}{2}(\text{m}\angle edf)$

Explanation:

Step1: Recall angle bisector definition

An angle bisector divides an angle into two equal - measure angles. So, if \(DF\) is the bisector of \(\angle CDE\), then \(m\angle CDF=m\angle EDF=\frac{1}{2}m\angle CDE\).

Step2: Analyze each option

• Option 1: \(\frac{1}{2}(m\angle CDF)=m\angle EDF\). Given \(m\angle CDF = 57^{\circ}\) and \(m\angle EDF=57^{\circ}\), \(\frac{1}{2}\times57^{\circ}=28.5^{\circ}

eq57^{\circ}\), so this option is wrong.

• Option 2: \(m\angle CDE = m\angle EDF\). \(m\angle CDE = 114^{\circ}\) and \(m\angle EDF = 57^{\circ}\), \(114^{\circ}

eq57^{\circ}\), so this option is wrong.

• Option 3: \(m\angle CDE=2(m\angle EDF)\). Substitute \(m\angle CDE = 114^{\circ}\) and \(m\angle EDF = 57^{\circ}\). \(2\times57^{\circ}=114^{\circ}\), so this equation holds.
• Option 4: \(m\angle CDF=\frac{1}{2}(m\angle EDF)\). \(m\angle CDF = 57^{\circ}\), \(\frac{1}{2}\times m\angle EDF=\frac{1}{2}\times57^{\circ}=28.5^{\circ}

eq57^{\circ}\), so this option is wrong.

Answer:

\( \boldsymbol{\text{m}\angle CDE = 2(\text{m}\angle EDF)} \) (the option with this statement)