Exercise 27 Page 526

Let's first note that $\overline{A C} ∥ \overline{B D}$ and $\overline{A C} ⊥ \overline{C D}$ should both apply. This means that $\overline{A C}$ has to be both parallel to $\overline{B D},$ and perpendicular to $\overline{C D} .$ Therefore, we can disregard figure B right away because it does not even include point $B .$ Let's consider the rest of the figures one at a time.

Diagram A

From this diagram we see that $\overline{A C}$ is perpendicular to $\overline{C D},$ which satisfies the second condition. Additionally, we see that $\overline{B D}$ is also perpendicular to $\overline{C D} .$ Therefore, according to the Lines Perpendicular to a Transversal Theorem, we can say that $\overline{A C} ∥ \overline{B D} .$

In diagram A is $\overline{A C} ∥ \overline{B D}$ and $\overline{A C} ⊥ \overline{C D} .$

Diagram C

Both $\overline{A B}$ and $\overline{C D}$ are perpendicular to $\overline{B D}$ which means they are parallel lines according to the Lines Perpendicular to a Transversal Theorem.

We also know that $\overline{A C} = \overline{B D},$ and since $\overline{B D}$ creates a $9 0^{\circ}$ angle with both $\overline{A B}$ and $\overline{C D},$ then we know that $\overline{A C}$ does too.

Now we see that $\overline{A C}$ is perpendicular to $\overline{C D},$ which satisfies the second condition. Additionally, we see that $\overline{B D}$ is also perpendicular to $\overline{C D} .$ Therefore, according to the Lines Perpendicular to a Transversal Theorem we can say that $\overline{A C}$ is parallel to $\overline{B D} .$

In diagram C is $\overline{A C} ∥ \overline{B D}$ and $\overline{A C} ⊥ \overline{C D} .$

Diagram D

This diagram has exactly the same configuration as figure A. Therefore, we can say that $\overline{A C}$ is perpendicular to $\overline{C D},$ which satisfies the second condition. Moreover, since we also see that $\overline{B D}$ is perpendicular to $\overline{C D},$ we have that $\overline{A C} ∥ \overline{B D}$ according to the Lines Perpendicular to a Transversal Theorem. Diagram D satisfies both conditions.

Diagram E

Again, this is the same configuration as diagram A, apart from the length of the horizontal sides. However, we are not considering the length of the sides. We are only looking at whether or not $\overline{A C}$ and $\overline{B D}$ are parallel and if $\overline{A C}$ is perpendicular to $\overline{C D} .$ Using the same logic as for diagram A, we can say that $\overline{A C} ∥ \overline{B D}$ and $\overline{A C} ⊥ \overline{C D} .$