Angles and Quadrilaterals

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GED Math › Angles and Quadrilaterals

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If the rectangle has a width of 5 and a length of 10, what is the area of the rectangle?

CORRECT

Explanation

Write the area for a rectangle.

Substitute the given dimensions.

The answer is:

In the figure below, find the measure of the largest angle.

$151^{\circ}$

CORRECT

$23^{\circ}$

$118^{\circ}$

$128^{\circ}$

Explanation

Recall that in a quadrilateral, the interior angles must add up to $360^{\circ}$ .

Thus, we can solve for :

Now, to find the largest angle, plug in the value of into each expression for each angle.

The largest angle is $151^{\circ}$ .

Given Quadrilateral , which of these statements would prove that it is a parallelogram?

I) $\angle A \cong \angle B$ and $\angle C \cong \angle D$

II) $\angle A \cong \angle C$ and $\angle B \cong \angle D$

III) $\angle A$ and $\angle B$ are supplementary and $\angle C$ and $\angle D$ are supplementary

Statement II only

CORRECT

Statement III only

Statement I only

Statement I, II, or III

Explanation

Statement I asserts that two pairs of consecutive angles are congruent. This does not prove that the figure is a parallelogram. For example, an isosceles trapezoid has two pairs of congruent base angles, which are consecutive.

Statement II asserts that both pairs of opposite angles are congruent. By a theorem of geometry, this proves the quadrilateral to be a parallelogram.

Statement III asserts that two pairs of consecutive angles are supplementary. While all parallelograms have this characteristic, trapezoids do as well, so this does not prove the figure a parallelogram.

The correct response is Statement II only.

In Rhombus , $m \angle R = 60 ^{\circ }$ . If $\overline{RO }$ is constructed, which of the following is true about $\Delta RHO$ ?

$\Delta RHO$ is obtuse and isosceles, but not equilateral

CORRECT

$\Delta RHO$ is acute and isosceles, but not equilateral

$\Delta RHO$ is right and isosceles, but not equilateral

$\Delta RHO$ is acute and equilateral

Explanation

The figure referenced is below.

Rhombus

The sides of a rhombus are congruent by definition, so $\overline{RH} \cong \overline{HO}$ , making $\Delta RHO$ isosceles (and possibly equilateral).

Also, consecutive angles of a rhombus are supplementary, as they are with all parallelograms, so

$m \angle H = 180^{\circ} - m \angle HRM = 180^{\circ} - 60 ^{\circ} = 120^{\circ}$ .

$\angle H$ , having measure greater than $90^{\circ}$ , is obtuse, making $\Delta RHO$ an obtuse triangle. Also, the triangle is not equilateral, since such a triangle must have three $60 ^{\circ}$ angles.

The correct response is that $\Delta RHO$ is obtuse and isosceles, but not equilateral.

You are given Parallelogram with . Which of the following statements, along with what you are given, would be enough to prove that Parallelogram is a rectangle?

I) $m \angle A = 90 ^{\circ }$

II) $m \angle B = 90 ^{\circ }$

III) $m \angle C = 90 ^{\circ }$

Statement I, II, or III

CORRECT

Statement III only

Statement II only

Statement I only

Explanation

A rectangle is defined as a parallelogram with four right, or $90 ^{\circ}$ , angles.

Since opposite angles of a paralellogram are congruent, if one angle measures $90 ^{\circ}$ , so does its opposite. Since consecutive angles of a paralellogram are supplementary - that is, their degree measures total $180 ^{\circ}$ - if one angle measures $90 ^{\circ}$ , then both of the neighboring angles measure $180 ^{\circ} - 90 ^{\circ} = 90 ^{\circ}$ .

In short, in a parallelogram, if one angle is right, all are right and the parallelogram is a rectangle. All three statements assert that one angle is right, so from any one, it follows that the figure is a rectangle. The correct response is Statements I, II, or III.

Note that the sidelengths are irrelevant.

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