Question

in parallelogram pqrs, if angle p = 70°, what is the measure of angle r?
a. 70°
b. 110°
c. 130°
d. 90°

in a parallelogram, which of the following relationships is always true?
a. the diagonals are congruent.
b. all angles are equal.
c. the sum of the measures of any two consecutive angles is 180°.
d. opposite angles are supplementary.

in rectangle hijk, hi = 4y + 6 and jk = 5y - 9. if opposite sides are congruent, what is the value of y?
a. 15
b. 9
c. 12
d. 6

in a parallelogram, the measure of two consecutive angles are 3x + 10° and 2x + 5°. what is the value of x?
a. x = 30
b. x = 25
c. x = 33
d. x = 20

an interior designer is creating a quadrilateral coffee table with vertices at a(0,0), b(6,2), c(8, - 2), and d(2, - 4). which property can prove that the table is a parallelogram?
a. opposite sides are congruent.
b. diagonals are congruent.
c. all angles are right angles.
d. opposite sides are perpendicular.

in parallelogram pqrs, if diagonal pr is bisected at point o and po = 3x + 4, while or = 5x - 2, what is the value of x?
a. 1
b. 4
c. 2
d. 3

Explanation:

Step1: Recall parallelogram angle - property

In a parallelogram, opposite angles are equal. In parallelogram $PQRS$, $\angle P$ and $\angle R$ are opposite angles.

Step2: Find the measure of $\angle R$

Since $\angle P = 70^{\circ}$ and $\angle P=\angle R$ (opposite - angles of a parallelogram), then $\angle R = 70^{\circ}$.

Step1: Recall parallelogram properties

The properties of a parallelogram are: 1. Opposite sides are parallel and congruent. 2. Opposite angles are equal. 3. The sum of any two consecutive angles is $180^{\circ}$. 4. Diagonals bisect each other.

Step2: Analyze each option

• Option a: Diagonals are congruent only in special parallelograms (rectangles, squares).
• Option b: All angles are equal only in rectangles and squares.
• Option c: The sum of the measures of any two consecutive angles in a parallelogram is $180^{\circ}$ because adjacent - sides are parallel and same - side interior angles are supplementary.
• Option d: Opposite angles are equal, not supplementary.

Step1: Use the property of opposite sides of a rectangle

In rectangle $HIJK$, opposite sides are congruent. So, $HI = JK$.

Step2: Set up the equation

We have the equation $4y + 6=5y−9$.

Step3: Solve for $y$

Subtract $4y$ from both sides: $4y + 6-4y=5y−9 - 4y$, which gives $6=y - 9$. Then add 9 to both sides: $y=6 + 9=15$.

Step1: Use the property of consecutive angles in a parallelogram

The sum of two consecutive angles in a parallelogram is $180^{\circ}$. So, $(3x + 10)+(2x + 5)=180$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $3x+2x+10 + 5=180$, which gives $5x+15 = 180$.

Step3: Solve for $x$

Subtract 15 from both sides: $5x=180 - 15=165$. Then divide both sides by 5: $x=\frac{165}{5}=33$.

Answer:

a. $70^{\circ}$