2. If $$ a, b, c $$, and $$ d $$ are coplanar lines and $$ a / / b, b \perp c $$, and $$ c / / d $$, then which statement must be true? Hint: Draw a picture to help you solve it. $$ d \perp c $$ $$ d \perp a $$ $$ d / / b $$ $$ c / / a $$

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UpStudy AI · Accepted Answer

1. We are given:
   - $$ a \parallel b $$
   - $$ b \perp c $$
   - $$ c \parallel d $$
   
2. Since $$ a \parallel b $$, $$ a $$ and $$ b $$ have the same direction. This implies the angle between $$ a $$ and any other line is the same as the angle between $$ b $$ and that line.

3. Because $$ b \perp c $$, the angle between $$ b $$ and $$ c $$ is $$ 90^\circ $$. Therefore, the angle between $$ a $$ and $$ c $$ is also $$ 90^\circ $$. Hence:
   $$
   a \perp c
   $$

4. Given that $$ c \parallel d $$, $$ d $$ has the same direction as $$ c $$. Therefore, the angle between $$ a $$ and $$ d $$ is the same as the angle between $$ a $$ and $$ c $$. Thus:
   $$
   a \perp d
   $$
   which can also be written as:
   $$
   d \perp a
   $$

5. Verifying the other options:
   - $$ d \perp c $$ is false because $$ d \parallel c $$, not perpendicular.
   - $$ d \parallel b $$ is false because $$ b \perp c $$ and $$ d \parallel c $$.
   - $$ c \parallel a $$ is false because we already established $$ a \perp c $$.

Thus, the statement that must be true is:
$$
d \perp a
$$
$$ d \perp a $$

If , and are coplanar lines and , and , then which statement must be true?
Hint: Draw a picture to help you solve it.

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If , and are coplanar lines and , and , then which statement must be true? Hint: Draw a picture to help you solve it.