Question

The angle measures of quadrilateral RSTU are shown.

m∠R = (2x)°

m∠S = (3x – 35)°

m∠T = (x + 35)°

The measure of angle U is unknown. Can quadrilateral RSTU be a parallelogram?

A. Yes, opposite angles R and T can be made equal to each other if x = 35.

B. Yes, consecutive angles R and S can be made equal to each other if x = 35.

C. No, if x = 35, all three given angles measure 70°. The fourth angle would measure 150°.

D. No, if x = 35, the three given angle measures make it impossible for the figure to be a quadrilateral.

Answer 1

The sum of the measures of all interior angles in a quadrilateral is always equal to 360°.

If m∠R = (2x)°, m∠S = (3x – 35)° and m∠T = (x + 35)°, then the measure of the angle U is

m∠U=360°-(2x)°-(3x-35)°-(x+35)°=360°-(2x)°-(3x)°+35°-x°-35°=360°-(6x)°.

If quadrilateral RSTU is parallelogram, then

• m∠R=m∠T;
• m∠S=m∠U;
• m∠R+m∠S=m∠S+m∠T=180°.

Check this conditions: 2x=x+35 ⇒ x=35°.

If x=35°, then

• m∠R=(2·35)°=70°;
• m∠S=(3·35-35)°=70°;
• m∠T=(35+35)°=70°;
• m∠U=360°-(6·35)°=150°.

You get quadrilateral RSTU that cannot be parallelogram, because 70°+70°≠180°.

Answer:  strictly correct choice is D (also choice C is correct)

Answer 2

Prallelogram has 2 pairs of opposite sides as parallel. The quadrilateral RSTU can not be a parallelogram.

What is parallelogram?

A parallelogram is a quadrilateral whose opposite sides are of equal length and the pair of opposite sides are parallel to each other.

We know that the sum of all the angles of a quadrilateral is 360°, therefore, in quadrilateral RSTU, the measure of ∠U can be written as,

∠R + ∠S + ∠T + ∠U = 360°

Substitute the values, ∠R = (2x)°, ∠S = (3x – 35)°, and ∠T = (x + 35)°,

(2x) + (3x – 35) + (x + 35) + ∠U = 360°

2x + 3x - 35 + x +35 + ∠U = 360°

∠U = 360° - 6x

We know that for a quadrilateral to be a parallelogram the opposite sides must be equal, therefore,

∠R = ∠T

∠S = ∠U

Let's substitute the values and check,

∠R = ∠T

$\angle R = \angle T\\2x = x +35\\2x-x = 35\\x =35$

∠S = ∠U

$\angle S = \angle U\\3x-35 = 360 -6x\\3x+6x = 360+35\\x = 43.889$

As we can see that when the value of x is different in both the cases and if we put the value of x as 35, then all three given angles measure 70°. The fourth angle would measure 150°.

Hence, the quadrilateral RSTU can not be a parallelogram.

Learn more about Parallelogram:

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