Question

self-assessment
state the theorem you can use to show that the quadrilateral is a parallelogram.
5
diagram of a quadrilateral with two sides labeled 30 m
6.
diagram of a quadrilateral with sides labeled 7 in and 5 in
7.
diagram of a quadrilateral with angles labeled 65°, 115°, 115°, 65°

Explanation:

To determine the theorem to show each quadrilateral is a parallelogram, we analyze each:

Problem 5 (Quadrilateral with two sides of 30 m)

A parallelogram has both pairs of opposite sides congruent (equal in length). Here, the quadrilateral has two opposite sides labeled \( 30 \, \text{m} \) (assuming the other two opposite sides are also congruent, though visually implied).

Theorem:

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Problem 6 (Quadrilateral with sides \( 7 \, \text{in}, 5 \, \text{in}, 7 \, \text{in}, 5 \, \text{in} \))

The side lengths are \( 7 \, \text{in}, 5 \, \text{in}, 7 \, \text{in}, 5 \, \text{in} \), so both pairs of opposite sides are congruent ( \( 7 = 7 \) and \( 5 = 5 \) ).

Theorem:

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Problem 7 (Quadrilateral with angles \( 65^\circ, 115^\circ, 115^\circ, 65^\circ \))

A parallelogram has both pairs of opposite angles congruent and consecutive angles supplementary (sum to \( 180^\circ \)). Here:

• Opposite angles: \( 65^\circ = 65^\circ \) and \( 115^\circ = 115^\circ \).
• Consecutive angles: \( 65^\circ + 115^\circ = 180^\circ \) (supplementary).

Theorem:

If both pairs of opposite angles of a quadrilateral are congruent, or if consecutive angles are supplementary, then the quadrilateral is a parallelogram. (Either theorem applies here.)

Final Answers:

• Problem 5: Use the theorem: If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.
• Problem 6: Use the theorem: If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.
• Problem 7: Use the theorem: If both pairs of opposite angles are congruent (or consecutive angles are supplementary), the quadrilateral is a parallelogram.

Answer:

A parallelogram has both pairs of opposite angles congruent and consecutive angles supplementary (sum to \( 180^\circ \)). Here:

• Opposite angles: \( 65^\circ = 65^\circ \) and \( 115^\circ = 115^\circ \).
• Consecutive angles: \( 65^\circ + 115^\circ = 180^\circ \) (supplementary).

Theorem:

If both pairs of opposite angles of a quadrilateral are congruent, or if consecutive angles are supplementary, then the quadrilateral is a parallelogram. (Either theorem applies here.)

Final Answers:

• Problem 5: Use the theorem: If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.
• Problem 6: Use the theorem: If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.
• Problem 7: Use the theorem: If both pairs of opposite angles are congruent (or consecutive angles are supplementary), the quadrilateral is a parallelogram.