Question

4
\\(\overline{cr}\\) and \\(\overline{ds}\\) are perpendiculars dropped from \\(\overleftrightarrow{ab}\\) to \\(\overleftrightarrow{pq}\\), and \\(\overleftrightarrow{ab}\\) is perpendicular to \\(\overline{cr}\\) and \\(\overline{ds}\\). if \\(cr = ds\\), which statement must be true?
a diagram with two horizontal lines, \\(\overleftrightarrow{ab}\\) (top) with points a, c, d, b and \\(\overleftrightarrow{pq}\\) (bottom) with points p, r, s, q. \\(\overline{cr}\\) and \\(\overline{ds}\\) are vertical segments with right angles at r and s respectively.
a. \\( m\angle rcd = m\angle sdb \div 2 \\)
b. \\( m\angle rcd = m\angle acd \\)
c. \\( m\angle rcd = m\angle acd \div 2 \\)
d. \\( m\angle rcd = m\angle acd \div 3 \\)
e. \\( m\angle rcd = m\angle acd \times 2 \\)

Explanation:

Step1: Identify quadrilateral CRSD

Since $\overline{CR} \perp \overleftrightarrow{PQ}$, $\overline{DS} \perp \overleftrightarrow{PQ}$, and $\overleftrightarrow{AB} \perp \overline{CR}$, all angles in quadrilateral $CRSD$ are right angles. Also, $CR=DS$, so $CRSD$ is a rectangle. Thus, $\overline{CD} \parallel \overline{RS}$ and $\overline{CR} = \overline{DS}$.

Step2: Analyze $\angle RCD$ and $\angle ACD$

$\angle ACD$ is a straight angle on line $\overleftrightarrow{AB}$, so $m\angle ACD = 180^\circ$. $\angle RCD$ is a right angle (since $\overline{CR} \perp \overleftrightarrow{AB}$), so $m\angle RCD = 90^\circ$.

Step3: Relate the two angles

Calculate the ratio: $\frac{m\angle ACD}{2} = \frac{180^\circ}{2} = 90^\circ = m\angle RCD$.

Answer:

C. $m\angle RCD = m\angle ACD \div 2$