Question

Write paragraph proofs. (In this book a star designates an exercise that is unusually difficult.)

Given: $\overline{\mathit{A}\mathit{E}}\mathbf{\Vert }\overline{\mathit{B}\mathit{D}}\mathbf{;}\mathbf{}\overline{\mathit{B}\mathit{C}}\mathbf{\Vert }\overline{\mathit{A}\mathit{D}}$

$\overline{\mathit{A}\mathit{E}}\mathbf{\cong }\overline{\mathit{B}\mathit{C}}\mathbf{;}\mathbf{}\overline{\mathit{A}\mathit{D}}\mathbf{\approx }\overline{\mathit{B}\mathit{D}}$

Prove:

a.$\overline{\mathit{A}\mathit{C}}\mathbf{\cong }\overline{\mathit{B}\mathit{E}}$

b.$\overline{\mathit{E}\mathit{C}}\mathbf{\Vert }\overline{\mathit{A}\mathit{B}}$

Short answer

a. By using regular trapezoid theorem it can proved that $\mathit{A}\mathit{C}\mathbf{\cong }\mathit{B}\mathit{E}$.

b. By using converse of regular trapezoid theorem it can proved that $\overline{\mathit{E}\mathit{C}}\mathbf{\Vert }\overline{\mathit{A}\mathit{B}}$.

Step-by-step solution

Part a. Step 1. State regular trapezoid theorem.

Regular trapezoid theorem states that the diagonals of a regular trapezoid are equal to each other.

Part a. Step 2. Observe quadrilaterals ABDE and ABCD

In quadrilateral$ABDE$

$AE$is parallel $BD$(given)

Therefore,$ABDE$ is a trapezoid.

$\Rightarrow AD=BE$(Diagonals of a regular trapezoid are equal) …. (i).

In quadrilateral$ABCD$

$BC$is parallel to $AD$(given)

$\therefore ABCD$is a trapezoid.

$\Rightarrow AC=BD$(Diagonals of a regular trapezoid are equal) …. (ii).

Part a. Step 3. State the conclusion.

Since it is given that $\overline{AD}\approx \overline{BD}$, therefore, from (i) and (ii) it can be concluded that

$AD=BD$and $AC=BE$.

$\therefore AC\cong BE$.

Hence Proved.

Part b. Step 1. Define converse of regular trapezoid theorem.

Converse of trapezoid theorem states that if diagonals of a trapezoid are equal to each other then it is a regular trapezoid.

Part b. Step 2. Observe quadrilaterals ABDE and ABCD

In quadrilateral$ADBE$

$AE$is parallel $BD$(given)

Therefore,$ADBE$is a trapezoid.

$\Rightarrow AD=BE$(Diagonals of a regular trapezoid are equal) …. (i).

In quadrilateral$ABCD$

$BC$is parallel to $AD$(given)

$\therefore ABCD$is a trapezoid.

$\Rightarrow AC=BD$(Diagonals of a regular trapezoid are equal) …. (ii).

Since it is given that $\overline{AD}\approx \overline{BD}$, therefore, from (i) and (ii) it can be concluded that

$AD=BD$and $AC=BE$.

Part b. Step 3. State the conclusion.

In quadrilateral $ABCE$,$AC=BE$ (proved above)

Therefore, $EC$is parallel to $AB$. (by the converse of regular trapezoid theorem)