given: $ \overleftrightarrow { a c } \parallel \overleftrightarrow { b d } $ $ \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } $, and $ \overleftrightarrow { a c } \perp \overleftrightarrow { c d } $.
prove: $ \angle p c q $ is complementary to $ \angle a b c $.

what is the missing step in the given proof?
$ \bigcirc $ a. $ \angle p q c $ and $ \angle a c p $ are supplementary by the linear pair theorem.
$ \bigcirc $ b. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle a c b \cong \angle p c q $.
$ \bigcirc $ c. $ \angle o c p \cong \angle b c d $ by the vertical angles theorem.
$ \bigcirc $ d. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle o c p \cong \angle a b c $.
$ \bigcirc $ e. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle o c a \cong \angle c b d $.

Question

given: $ \overleftrightarrow { a c } \parallel \overleftrightarrow { b d } $ $ \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } $, and $ \overleftrightarrow { a c } \perp \overleftrightarrow { c d } $.
prove: $ \angle p c q $ is complementary to $ \angle a b c $.

what is the missing step in the given proof?
$ \bigcirc $ a. $ \angle p q c $ and $ \angle a c p $ are supplementary by the linear pair theorem.
$ \bigcirc $ b. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle a c b \cong \angle p c q $.
$ \bigcirc $ c. $ \angle o c p \cong \angle b c d $ by the vertical angles theorem.
$ \bigcirc $ d. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle o c p \cong \angle a b c $.
$ \bigcirc $ e. for parallel lines cut by a transversal, corresponding angles are congruent, so $ \angle o c a \cong \angle c b d $.

Accepted Answer

# Explanation:
## Step1: Analyze each option
- Option A: $\angle PQC$ and $\angle ACP$ are not relevant to proving $\angle PCQ$ is complementary to $\angle ABC$.
- Option B: $\angle ACB$ and $\angle PCQ$ are not directly related to the given parallel lines and the angle - complement relationship.
- Option C: $\angle OCP$ and $\angle BCD$  have no connection to the proof of $\angle PCQ$ and $\angle ABC$ being complementary.
- Option D: Since $\overleftrightarrow{AC}\parallel\overleftrightarrow{BD}$ (given) and they are cut by transversal $\overleftrightarrow{AB}$, by the Corresponding Angles Theorem, $\angle OCA\cong\angle CBD$. This is not relevant to the proof of $\angle PCQ$ and $\angle ABC$ being complementary.
- Option E: Because $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$ (given) and they are cut by transversal $\overleftrightarrow{BC}$, by the Corresponding Angles Theorem, $\angle OCP\cong\angle ABC$. Also, since $\overleftrightarrow{AC}\perp\overleftrightarrow{CD}$, $\angle ACD = 90^{\circ}$. And $\angle ACD=\angle OCP+\angle PCQ$ (angle - addition postulate). So $\angle PCQ$ is complementary to $\angle OCP$. Since $\angle OCP\cong\angle ABC$, $\angle PCQ$ is complementary to $\angle ABC$.

# Answer:
C. $\angle OCP\cong\angle BCD$ by the Vertical Angles Theorem.

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Explanation:

Step1: Analyze each option

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