4. justify reasoning determine whether each statement is always, sometimes, or never true. you may 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 g h j k is a parallelogram, then $\overline{gh}$ is congruent to $\overline{jk}$.
in parallelogram a b c d, $\angle a$ is acute and $\angle c$ is obtuse.
in parallelogram m n p q, the diagonals $\overline{mp}$ and $\overline{nq}$ meet at r with $mr = 7$ cm and $rp = 5$ cm.

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# Explanation: ## 1. For the statement "If a parallelogram has a $30^{\circ}$ angle, then it also has a $150^{\circ}$ angle" - **Step1: Recall the properties of parallelogram angles** In a parallelogram, consecutive angles are supplementary (i.e., their sum is $180^{\circ}$). Let one angle be $\angle A = 30^{\circ}$. If $\angle B$ is a consecutive angle to $\angle A$, then $\angle A+\angle B = 180^{\circ}$. - **Step2: Solve for the consecutive angle** Substitute $\angle A = 30^{\circ}$ into the equation $\angle A+\angle B=180^{\circ}$. We get $30^{\circ}+\angle B = 180^{\circ}$, so $\angle B=180^{\circ}- 30^{\circ}=150^{\circ}$ ## 2. For the statement "If quadrilateral $GHJK$ is a parallelogram, then $\overline{GH}$ is congruent to $\overline{JK}$" - **Step1: Recall the property of parallelogram sides** In a parallelogram, opposite sides are congruent. If $GHJK$ is a parallelogram, then $GH$ and $JK$ are opposite sides. ## 3. For the statement "In parallelogram $ABCD$, $\angle A$ is acute and $\angle C$ is obtuse" - **Step1: Recall the property of parallelogram opposite angles** In a parallelogram, opposite angles are congruent. So $\angle A=\angle C$. An acute angle is an angle with measure between $0^{\circ}$ and $90^{\circ}$, and an obtuse angle is an angle with measure between $90^{\circ}$ and $180^{\circ}$. If $\angle A$ is acute ($\angle A<90^{\circ}$), then $\angle C=\angle A<90^{\circ}$ (acute), and if $\angle A$ is obtuse ($\angle A > 90^{\circ}$), then $\angle C=\angle A>90^{\circ}$ (obtuse) ## 4. For the statement "In parallelogram $MNPQ$, the diagonals $\overline{MP}$ and $\overline{NQ}$ meet at $R$ with $MR = 7\mathrm{cm}$ and $RP=5\mathrm{cm}$" - **Step1: Recall the property of parallelogram diagonals** In a parallelogram, diagonals bisect each other. So if the diagonals $MP$ and $NQ$ intersect at $R$, then $MR = RP$ # Answer: - "If a parallelogram has a $30^{\circ}$ angle, then it also has a $150^{\circ}$ angle": Always; because consecutive angles of a parallelogram are supplementary ($30 + 150=180$) - "If quadrilateral $GHJK$ is a parallelogram, then $\overline{GH}$ is congruent to $\overline{JK}$": Always; by the property that opposite sides of a parallelogram are congruent - "In parallelogram $ABCD$, $\angle A$ is acute and $\angle C$ is obtuse": Never; since opposite angles of a parallelogram are congruent - "In parallelogram $MNPQ$, the diagonals $\overline{MP}$ and $\overline{NQ}$ meet at $R$ with $MR = 7\mathrm{cm}$ and $RP = 5\mathrm{cm}$": Never; because diagonals of a parallelogram bisect each other ($MR$ should equal $RP$)

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Explanation:

1. For the statement "If a parallelogram has a \(30^{\circ}\) angle, then it also has a \(150^{\circ}\) angle"

2. For the statement "If quadrilateral \(GHJK\) is a parallelogram, then \(\overline{GH}\) is congruent to \(\overline{JK}\)"

3. For the statement "In parallelogram \(ABCD\), \(\angle A\) is acute and \(\angle C\) is obtuse"

4. For the statement "In parallelogram \(MNPQ\), the diagonals \(\overline{MP}\) and \(\overline{NQ}\) meet at \(R\) with \(MR = 7\mathrm{cm}\) and \(RP=5\mathrm{cm}\)"

Answer: