Question

1. ABCD is a parallelogram. Find the value of x.

A. x = 9
B. x = 22
C. x = 16.5
D. x = 123

2. What are the measurements of the angles of the parallelogram in the previous question?

A. Two of the angles measure 57°, and the other two measure 123°.
B. Two of the angles measure 83°, and the other two measure 97°.
C. Two of the angles measure 32°, and the other two measure 148°.
D. Two of the angles measure 84.5°, and the other two measure 95.5°.

Answer 1

Answer:

Step-by-step explanation:

1. Since, it is given that ABCD is a parallelogram, therefore opposite angles of parallelogram are equal, hence

∠B=∠D

⇒7x-31=5x+13

⇒2x=44

⇒x=22

Hence, option B is correct.

2. Since, we get x=22, therefore

∠B=7x-31=7(22)-31=154-31=123°

∠B=123°

and ∠D=5x+13=5(22)+13=123°

∠D=123°

Also, ∠A+∠D=180°(corresponding angles)

⇒∠A=180-123

⇒∠A=57°

Now, ∠A=∠C=57°(opposite angles of parallelogram)

thus, Two of the angles measure 57°, and the other two measure 123°.

Therefore, option A is correct.

Answer 2

You can use that a parallelogram has opposite angles congruent.

The value of x is

Option B: x = 22

What are the measurements of the angles of a parallelogram?

Opposite angles of a parallelogram are congruent. It is because if you draw a diagonal in a parallelogram, you can prove that both the triangles on either side of that diagonals are congruent by SSS congruency.

If ABCD is a parallelogram, then

$\angle A = \angle C\\\angle B = \angle D$

The adjacent angles of a parallelogram are supplementary, ie they add up to 180 degrees.

If ABCD is a parallelogram, then

$\angle A + \angle B = 180^\circ\\\angle A + \angle D = 180^\circ\\\angle B + \angle C = 180^\circ\\\angle C + \angle D = 180^\circ$

How to find the value of x?

Using that aforesaid fact about equality of two angles which are lying opposite to each other (those who face each other but are not near to each other(not adjacent), we have:

$\angle A = \angle C\\\angle B = \angle D$

Thus,

$\angle B = \angle D\\5x + 13 = 7x - 31 \text{\: (In degrees)}\\\\\text{Adding 31 - 2x on both sides}\\ \text{so that x related terms get on one side and constants get on other side}\\\\5x + 13 + 31 - 5x = 7x - 31 + 31 - 5x\\44 = 2x\\2x= 44\\\\\x = \dfrac{44}{2}\\\\x= 22$

Thus,

The value of x is

Option B: x = 22

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