Question

from \\( \overleftrightarrow { a b } \\) to \\( \overleftrightarrow { p q } \\), and \\( \overleftrightarrow { a b } \\) is perpendicular to \\( \overline { c r } \\) and \\( \overline { d s } \\). if \\( c r = d s \\), which statement must be true?

a. \\( m \angle r c d = m \angle s d b + 2 \\)
b. \\( m \angle r c d = m \angle a c d \\)
c. \\( m \angle r c d = m \angle a c d + 2 \\)
d. \\( m \angle r c d = m \angle a c d + 3 \\)
e. \\( m \angle r c d = m \angle a c d \times 2 \\)

Explanation:

Step1: Analyze the given information

Since \(\overleftrightarrow{AB}\parallel\overleftrightarrow{PQ}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{CR}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{DS}\), and \(CR = DS\).
We know that \(CR\parallel DS\) (both perpendicular to \(AB\)) and \(CR = DS\). So, \(CRSD\) is a parallelogram (a quadrilateral with one pair of parallel and equal sides is a parallelogram).
In a parallelogram \(CRSD\), \(CD\) is a diagonal.

Step2: Use the property of parallelogram

In parallelogram \(CRSD\), \(\angle RCD=\angle SDC\) (alternate - interior angles for \(CR\parallel DS\) and transversal \(CD\)).
Also, since \(CR\parallel DS\) and \(AB\) is a transversal, \(\angle ACD=\angle SDC\) (alternate - interior angles).

Answer:

B. \(m\angle RCD = m\angle ACD\)