Question

theorems
7.9 opposite sides parallel and congruent theorem
if one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
if \\(\overline{bc} \parallel \overline{ad}\\) and \\(\overline{bc} \cong \overline{ad}\\), then \\(abcd\\) is a parallelogram.
prove this theorem exercise 39, page 611

7.10 parallelogram diagonals converse
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
if \\(\overline{bd}\\) and \\(\overline{ac}\\) bisect each other, then \\(abcd\\) is a parallelogram.
prove this theorem exercise 40, page 611

image of a door labeled with vertices s, t, u, v

example 3 identifying a parallelogram
the doorway shown is part of a building. over time, the build sideways. explain how you know that \\(sv = tu\\).

solution
in the photograph, \\(\overline{st} \parallel \overline{uv}\\) and \\(\overline{st} \cong \overline{uv}\\). by the opposite and congruent theorem, quadrilateral \\(stuv\\) is a parallelog
parallelogram opposite sides theorem, you know that opp a parallelogram are congruent. so, \\(sv = tu\\).

elf-assessment 1 i dont understand yet. 2 i can do it with help. 3 i can do it on my

1. writing a quadrilateral has four congruent sides. is the quadrilateral a parallelogram? justify your answer.

1. different words, same question which is different? find \both\ answers.

• construct a quadrilateral with congruent opposite sides.
• construct a quadrilateral with one pair of parallel sides.
• construct a quadrilateral with congruent opposite angles.
• construct a quadrilateral with one pair of opposite sides parallel and congruent.

Explanation:

Problem 3: Writing - Quadrilateral with Four Congruent Sides as a Parallelogram

Step 1: Recall Parallelogram Definition/Properties

A parallelogram is a quadrilateral with both pairs of opposite sides parallel (and congruent, by the Parallelogram Opposite Sides Theorem). If a quadrilateral has four congruent sides, then opposite sides are congruent (since all sides are congruent, so \(AB = CD\) and \(BC = AD\) for quadrilateral \(ABCD\)).

Step 2: Apply the Opposite Sides Congruent Theorem

The theorem (Opposite Sides Congruent Theorem for Parallelograms, or its converse: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram) tells us that if \(AB \cong CD\) and \(BC \cong AD\), then \(ABCD\) is a parallelogram. Since four congruent sides imply both pairs of opposite sides are congruent, the quadrilateral satisfies this condition.

• Options 1, 3, and 4 all result in constructing a parallelogram (since their conditions define a parallelogram).
• Option 2 ("Construct a quadrilateral with one pair of parallel sides") constructs a trapezoid (which is not necessarily a parallelogram, as it only requires one pair of parallel sides, not two).

Answer:

Yes, the quadrilateral is a parallelogram. Justification: If a quadrilateral has four congruent sides, then both pairs of its opposite sides are congruent (e.g., for quadrilateral \(ABCD\), \(AB = CD\) and \(BC = AD\) because all sides \(AB = BC = CD = AD\)). By the theorem that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram, this quadrilateral is a parallelogram.

Problem 4: Different Words, Same Question

To determine which is different, we analyze each option:

1. "Construct a quadrilateral with congruent opposite sides."

By the Opposite Sides Congruent Theorem (converse), a quadrilateral with congruent opposite sides is a parallelogram (so this constructs a parallelogram, or a quadrilateral that is a parallelogram).

1. "Construct a quadrilateral with one pair of parallel sides."

A quadrilateral with only one pair of parallel sides is a trapezoid (not necessarily a parallelogram, since parallelograms require both pairs of opposite sides parallel).

1. "Construct a quadrilateral with congruent opposite angles."

By the theorem, if a quadrilateral has congruent opposite angles, it is a parallelogram (since consecutive angles in a parallelogram are supplementary, and opposite angles are equal).

1. "Construct a quadrilateral with one pair of opposite sides parallel and congruent."

By the Opposite Sides Parallel and Congruent Theorem, this quadrilateral is a parallelogram.