Question

1. if rectangle abcd is similar to rectangle mnop, which of the following must be true?

\\(\overline{cd}\cong\overline{op})\
\\(\angle c\cong\angle o\\)\
\\(\angle b\cong\angle m\\)\
\\(\overline{bc}\cong\overline{no})

Explanation:

Step1: Recall property of similar rectangles

For similar rectangles, corresponding angles are congruent and corresponding sides are in proportion.

Step2: Analyze angle - side correspondences

In similar rectangles \(ABCD\) and \(MNOP\), \(\angle A\) corresponds to \(\angle M\), \(\angle B\) corresponds to \(\angle N\), \(\angle C\) corresponds to \(\angle O\), and \(\angle D\) corresponds to \(\angle P\). Also, \(\overline{AB}\) corresponds to \(\overline{MN}\), \(\overline{BC}\) corresponds to \(\overline{NO}\), \(\overline{CD}\) corresponds to \(\overline{OP}\), and \(\overline{DA}\) corresponds to \(\overline{PM}\).

Step3: Evaluate each option

• Option 1: \(\overline{CD}\cong\overline{OP}\) is wrong as for similar rectangles, sides are in proportion not necessarily congruent.
• Option 2: Since \(\angle C\) and \(\angle O\) are corresponding angles in similar rectangles \(ABCD\) and \(MNOP\), \(\angle C\cong\angle O\).
• Option 3: \(\angle B\) corresponds to \(\angle N\), not \(\angle M\), so \(\angle B\cong\angle M\) is wrong.
• Option 4: \(\overline{BC}\) corresponds to \(\overline{NO}\), but they are in proportion not necessarily congruent.

Answer:

\(\angle C\cong\angle O\)