abcd is a quadrilateral in which opposite angles have the same measure, as indicated in the following figure. what kind of quadrilateral is abcd? justify your answer
complete the sentences below
because the sum of the interior angles is $\square^\circ$, $\angle a$ and $\angle d$ are dropdown also $\angle d$ and $\angle c$ are dropdown if we extend the side dropdown to form an alternate interior angle to $\angle c$, we see that these angles are dropdown and thus, dropdown is parallel to ad. similarly, ab is parallel to dc. thus, abcd is a dropdown (type whole numbers.)

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Accepted Answer

# Explanation:
## Step1: Recall interior angles of quadrilateral
The sum of the interior angles of any quadrilateral is calculated by the formula $(n - 2)\times180^{\circ}$, where $n = 4$ for a quadrilateral. So $(4 - 2)\times180^{\circ}= 360^{\circ}$.
## Step2: Analyze angle relationships
In a quadrilateral $ABCD$ with opposite angles equal, we use the property of parallel lines and transversals. $\angle A$ and $\angle D$ are supplementary (since consecutive angles in a quadrilateral with opposite angles equal and sum of interior angles $360^{\circ}$, if $\angle A=\angle C$ and $\angle B = \angle D$, then $\angle A+\angle D+\angle B+\angle C=360^{\circ}$ implies $\angle A+\angle D = 180^{\circ}$). Also, $\angle D$ and $\angle C$ are supplementary? Wait, no, let's correct. Wait, when we extend a side, say $BC$ (or other side), alternate interior angles: if we consider $AB$ and $DC$ with transversal $AD$, $\angle A$ and $\angle D$ are same - side interior angles? Wait, no, the key property is: In a quadrilateral, if opposite angles are equal, then it's a parallelogram. The sum of interior angles of a quadrilateral is $360^{\circ}$. For $\angle A$ and $\angle D$, since $\angle A=\angle C$ and $\angle B=\angle D$, $\angle A+\angle D+\angle B+\angle C = 2(\angle A+\angle D)=360^{\circ}$, so $\angle A+\angle D = 180^{\circ}$, which means $AB\parallel DC$ (same - side interior angles supplementary). Similarly, $\angle A+\angle B=180^{\circ}$ (since $\angle B = \angle D$ and $\angle A+\angle D = 180^{\circ}$), so $AD\parallel BC$. So $ABCD$ is a parallelogram.

Filling the blanks:
- The sum of the interior angles is $360^{\circ}$.
- $\angle A$ and $\angle D$ are supplementary.
- $\angle D$ and $\angle C$ are supplementary (wait, no, maybe alternate interior angles? Wait, if we extend $BC$ to form a transversal with $AD$, then $\angle D$ and $\angle C$ (wait, maybe the correct approach is: Since $\angle A=\angle C$ and $\angle B=\angle D$, and sum of interior angles is $360^{\circ}$, so consecutive angles are supplementary. So $\angle A+\angle D = 180^{\circ}$, so $AB\parallel DC$ (same - side interior angles supplementary). Similarly, $\angle B+\angle C=180^{\circ}$, so $AD\parallel BC$. So the quadrilateral with both pairs of opposite sides parallel is a parallelogram.

So the first blank: $360$ (since sum of interior angles of quadrilateral is $(4 - 2)\times180=360$).

$\angle A$ and $\angle D$ are supplementary (because $\angle A+\angle D+\angle B+\angle C = 360$ and $\angle A=\angle C$, $\angle B=\angle D$, so $2(\angle A+\angle D)=360$ implies $\angle A+\angle D = 180$, so supplementary).

$\angle D$ and $\angle C$: Wait, maybe the other pair. If we consider $AD$ and $BC$ with transversal $DC$, then $\angle D$ and $\angle C$ are same - side interior angles? No, maybe the problem is structured as:

Because the sum of the interior angles is $360^{\circ}$, $\angle A$ and $\angle D$ are supplementary. Also, $\angle D$ and $\angle C$ are supplementary? Wait, no, let's think again. The key is that in a quadrilateral with opposite angles equal, it's a parallelogram. The sum of interior angles of a quadrilateral is $360^{\circ}$. For the parallel sides: if $\angle A+\angle D = 180^{\circ}$, then $AB\parallel DC$ (same - side interior angles supplementary). Similarly, $\angle B+\angle C=180^{\circ}$, so $AD\parallel BC$. So the quadrilateral $ABCD$ is a parallelogram.

So filling the first blank: $360$ (sum of interior angles of quadrilateral).

$\angle A$ and $\angle D$ are supplementary (since their sum is $180^{\circ}$ as shown above).

If we extend the side $BC$ (for example) to form a transversal, and $\angle D$ and $\angle C$ (wait, maybe the alternate interior angles: if we have $AB\parallel DC$ and transversal $AD$, then $\angle A$ and $\angle D$ are same - side interior angles. Wait, maybe the problem is using the property that if consecutive angles are supplementary, then sides are parallel.

So to summarize:

1. Sum of interior angles of quadrilateral: $(4 - 2)\times180 = 360^{\circ}$.
2. $\angle A$ and $\angle D$ are supplementary (since $\angle A+\angle D+\angle B+\angle C=360$ and $\angle A=\angle C$, $\angle B=\angle D$, so $2(\angle A+\angle D)=360$ implies $\angle A+\angle D = 180$).
3. $\angle D$ and $\angle C$: Wait, maybe the other pair. If $\angle B=\angle D$ and $\angle A=\angle C$, then $\angle B+\angle C=180$ as well, so $\angle D+\angle C = 180$ (since $\angle B=\angle D$). So $\angle D$ and $\angle C$ are supplementary.
4. If we extend the side $BC$ (to form a transversal with $AD$), and since $\angle D$ and $\angle C$ are supplementary? No, alternate interior angles: if we have $AB\parallel DC$ and transversal $BC$, then $\angle B$ and $\angle C$ are same - side interior angles. Wait, maybe the problem is leading to the conclusion that $BC\parallel AD$ and $AB\parallel DC$, so $ABCD$ is a parallelogram.

So the first blank: $360$.

$\angle A$ and $\angle D$ are supplementary.

$\angle D$ and $\angle C$ are supplementary.

If we extend the side $BC$ (for example) to form an alternate interior angle with $\angle C$, we see that these angles are equal (wait, no, maybe the correct flow is: Since $\angle A=\angle C$ and $\angle B=\angle D$, and sum of interior angles is $360$, so consecutive angles are supplementary. So $AB\parallel DC$ (because $\angle A+\angle D = 180$) and $AD\parallel BC$ (because $\angle B+\angle C=180$). Thus, $ABCD$ is a parallelogram.

# Answer:
- The sum of the interior angles is $\boldsymbol{360}$°.
- $\angle A$ and $\angle D$ are supplementary.
- $\angle D$ and $\angle C$ are supplementary.
- If we extend the side $BC$ (or other appropriate side) to form an alternate interior angle to $\angle C$, we see that these angles are equal (or supplementary, depending on the exact construction), and thus, $BC$ is parallel to $AD$. Similarly, $AB$ is parallel to $DC$. Thus, $ABCD$ is a parallelogram.

(For the blanks in the problem, filling the first blank with $360$, the relationship between $\angle A$ and $\angle D$ as supplementary, between $\angle D$ and $\angle C$ as supplementary, the side to extend as $BC$ (or $AB$ depending on the figure), and the quadrilateral as a parallelogram.)

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Explanation:

Step1: Recall interior angles of quadrilateral

Step2: Analyze angle relationships

Answer: