Question

1. justify reasoning determine whether each statement is always, sometimes, or never true. you may draw a picture of each situation to help you. explain your reasoning. the first one is done for you.

if quadrilateral \\(rstu\\) is a parallelogram, then \\(\overline{rs} \cong \overline{st}\\).

sometimes; opposite sides of a parallelogram are congruent, but consecutive sides may or may not be congruent

if a parallelogram has a \\(30^\circ\\) angle, then it also has a \\(150^\circ\\) angle.

if quadrilateral \\(ghjk\\) is a parallelogram, then \\(\overline{gh}\\) is congruent to \\(\overline{jk}\\).

in parallelogram \\(abcd\\), \\(\angle a\\) is acute and \\(\angle c\\) is obtuse.

in parallelogram \\(mnpq\\), the diagonals \\(\overline{mp}\\) and \\(\overline{nq}\\) meet at \\(r\\) with \\(mr = 7\text{ cm}\\) and \\(rp = 5\text{ cm}\\).

Explanation:

Analyze the first statement

Using the Parallelogram Theorems and Quadrilateral Properties knowledge points

• Statement: "If a parallelogram has a \(30^\circ\) angle, then it also has a \(150^\circ\) angle."
• Consecutive angles in any parallelogram are supplementary.
• If one angle is \(30^\circ\), the consecutive angle must be \(180^\circ - 30^\circ = 150^\circ\).
• Therefore, this statement is always true.

Analyze the second statement

Using the Parallelogram Theorems and Quadrilateral Properties knowledge points

• Statement: "If quadrilateral \(GHJK\) is a parallelogram, then \(\overline{GH}\) is congruent to \(\overline{JK}\)."
• In a parallelogram named in order \(G-H-J-K\), the opposite sides are \(\overline{GH}\) and \(\overline{JK}\).
• Opposite sides of a parallelogram are always congruent.
• Therefore, this statement is always true.

Analyze the third statement

Using the Parallelogram Theorems and Quadrilateral Properties knowledge points

• Statement: "In parallelogram \(ABCD\), \(\angle A\) is acute and \(\angle C\) is obtuse."
• In parallelogram \(ABCD\), \(\angle A\) and \(\angle C\) are opposite angles.
• Opposite angles of a parallelogram must be congruent.
• An angle cannot be both acute (less than \(90^\circ\)) and obtuse (greater than \(90^\circ\)) while remaining congruent.
• Therefore, this statement is never true.

Analyze the fourth statement

Using the Parallelogram Theorems and Quadrilateral Properties knowledge points

• Statement: "In parallelogram \(MNPQ\), the diagonals \(\overline{MP}\) and \(\overline{NQ}\) meet at \(R\) with \(MR = 7\text{ cm}\) and \(RP = 5\text{ cm}\)."
• The diagonals of a parallelogram bisect each other.
• The intersection point \(R\) must divide diagonal \(\overline{MP}\) into two equal segments, meaning \(MR\) must equal \(RP\).
• Since \(7\text{ cm}

eq 5\text{ cm}\), this configuration is impossible.

• Therefore, this statement is never true.

Answer:

Question 1

• Statement: If a parallelogram has a \(30^\circ\) angle, then it also has a \(150^\circ\) angle.
• Answer: Always; consecutive angles of a parallelogram are supplementary, so \(180^\circ - 30^\circ = 150^\circ\).

Question 2

• Statement: If quadrilateral \(GHJK\) is a parallelogram, then \(\overline{GH}\) is congruent to \(\overline{JK}\).
• Answer: Always; opposite sides of a parallelogram are always congruent.

Question 3

• Statement: In parallelogram \(ABCD\), \(\angle A\) is acute and \(\angle C\) is obtuse.
• Answer: Never; opposite angles of a parallelogram must be congruent, so one cannot be acute while the other is obtuse.

Question 4

• Statement: In parallelogram \(MNPQ\), the diagonals \(\overline{MP}\) and \(\overline{NQ}\) meet at \(R\) with \(MR = 7\text{ cm}\) and \(RP = 5\text{ cm}\).
• Answer: Never; the diagonals of a parallelogram bisect each other, which requires \(MR = RP\).