Question

given: in parallelogram abcd,
not drawn to scale.
what is the measurement of ∠ b?
question 4
6.67 pts
let ac and bd be two segments that bisect one another. which of the following is true about quadrilateral abcd?
○ quadrilateral abcd must be a rhombus and could be a rectangle.
○ quadrilateral abcd must be an isosceles trapezoid.
○ quadrilateral abcd must be a parallelogram and could be a rectangle or a rhombus.
○ quadrilateral abcd must be a rectangle and could be a rhombus.

Explanation:

First Problem (Measurement of ∠B):

Step1: Set up equations for opposite angles

In a parallelogram, opposite angles are equal:
$7y + 13 = 10y - 32$ and $106 - 2x = 3x - 4$

Step2: Solve for y

$7y + 13 = 10y - 32$
$32 + 13 = 10y - 7y$
$45 = 3y$
$y = 15$

Step3: Solve for x

$106 - 2x = 3x - 4$
$106 + 4 = 3x + 2x$
$110 = 5x$
$x = 22$

Step4: Calculate ∠B

$\angle B = 106 - 2x = 106 - 2(22) = 106 - 44 = 62$

By the parallelogram theorem, if diagonals bisect each other, the quadrilateral is a parallelogram. A parallelogram can be a rectangle (if diagonals are equal) or a rhombus (if diagonals are perpendicular), so it must be a parallelogram and could be either.

Answer:

62

Second Problem (Question 4):