Quadrilaterals - Intermediate Geometry

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Question

A parallelogram contains 2 angles measuring 135 and 45. What are the measures of the other 2 angles?

Answer

Parallelograms have angles totalling 360 degrees, but also have matching pairs of angles at the ends of diagonals. Therefore the 2 additional angles must match the 2 given in the question.

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Question

Using the above rhombus, find the measurement of angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

Thus, the solution is:

$\measuredangle D=180$

$\measuredangle D=180-110=70$

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Question

Using the above rhombus, find the measurement of angle .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

Thus, $\measuredangle C=\measuredangle A$

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Question

Using the above rhombus, find the measurement of angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$\measuredangle H+\measuredangle G=180^\circ$
$47+\measuredangle G=180^\circ$
$\measuredangle G=180-47=133^\circ$

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Question

Using the above rhombus, find the sum of angle and angle .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$\measuredangle H=47^\circ$

$\measuredangle F=47^\circ$

$47^\circ+47^\circ=94^\circ$

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Question

Given that the measurement of angle degrees, find the sum of angle and angle

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

The solution to this problem is:

$\measuredangle B+\measuredangle C=180^\circ$

$52+\measuredangle C=180^\circ$

$C= 180-52=128^\circ$

$\measuredangle C=\measuredangle A$

Therefore,

$128+128=256^\circ$

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Question

In the above rhombus, angle has a measurement of degrees. Find the sum of angles and

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees)--i.e. angles degrees.

The solution to this problem is:

$\measuredangle D=\measuredangle B$

Thus,

$52+52=104^\circ$

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Question

Using the parallelogram above, find the measurement of angle

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angle and are supplementary the solution is:

$180-99=81^\circ$

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Question

Using the parallelogram above, find the sum of angles and .

Answer

A rhombus must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

The first step to solving this problem is to find the measurement of angle . Since angle is a supplementary angle to angle , angle

$180-99=81^\circ$

Since, angle and are opposite interior angles they must be equivalent.

Thus, the final solution is:

$81+81=162^\circ$

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Question

Using the parallelogram above, find the sum of angles and .

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angles and are opposite interior angles, they must be equivalent.

Therefore, the solution is:

$99+99=198^\circ$

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Question

In the parallelogram shown above, angle is degrees. Find the measure of angle

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angles and are opposite interior angles, thus they must be equivalent.

$\measuredangle E=72^\circ$ , therefore $\measuredangle G=72^\circ$

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Question

In the parallelogram shown above, angle is degrees. Find the sum of angles and .

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees. And, the adjacent interior angles must be supplementary angles (sum of degrees).

Thus, the solution is:

$180-72=108^\circ$

Since both angles and equal There sum must equal $108+108=216^\circ$

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Question

Using the parallelogram above, find the sum of angles and

Answer

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal degrees.

Also, the adjacent interior angles must be supplementary angles (sum of degrees).

Since, angles and are adjacent to each other they must be supplementary angles.

Thus, the sum of these two angles must equal degrees.

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Question

A paralellogram as two angles that are 65 degrees and 115 degrees respectively. What are the other two angles in the paralellogram?

Answer

This question is very simple to answer if you remember that ALL paralellograms have two pairs of equal and opposite angles, and that the four angles in any quadrilateral MUST add up to 360 degrees.

Because the angles given are different, we know that they are supplementary and the other two missing angles MUST be the same.

$65^{\circ} + 115^{\circ} + 65^{\circ} + 115^{\circ} = 360^{\circ}$

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Question

Given: Rectangle with diagonals $\overline{AC}$ and $\overline{BD}$ intersecting at point .

True or false: $\angle AXB$ must be a right angle.

Answer

The diagonals of a parallelogram are perpendicular - and, consequently, $\angle AXB$ is a right angle. - if and only if the parallelogram is a rhombus, a figure with four sides of equal length. Not all rectangles have four congruent sides. Therefore, $\angle AXB$ need not be a right angle.

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Question

Given: Regular Pentagon with center . Construct segments $\overline{CP}$ and $\overline{CN}$ to form Quadrilateral .

True or false: Quadrilateral is a parallelogram.

Answer

Below is regular Pentagon with center , a segment drawn from to each vertex - that is, each of its radii drawn.

The measure of each angle of a regular pentagon can be calculated by setting equal to 5 in the formula

$\frac{(N-2)180^{\circ }}{N}$

and evaluating:

$\frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }$

Specifically,

$m \angle PEN = 108 ^{\circ }$

By symmetry, each radius bisects one of these angles. Specifically,

$m \angle CPE = \frac{1}{2} m \angle APE = \frac{1}{2} \cdot 108^{\circ } = 54 ^{\circ }$

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram are supplementary - that is, their measures total $180^{\circ }$ . However,

$m \angle PEN + m \angle CPE = 108 ^{\circ } + 54 ^{\circ } = 162^{\circ }$ ,

violating these conditions. Therefore, Quadrilateral is not a parallelogram.

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Question

Given: Parallelogram such that $m \angle A = 90^{\circ }$ .

True or false: Parallelogram must be a rectangle.

Answer

A rectangle is a parallelogram with four right angles.

Consecutive angles of a parallelogram are supplementary. If one angle of a parallelogram is given to be right, then its neighboring angles, being supplementary to a right angle, are right as well; also, opposite angles of a parallelogram are congruent, so the opposite angle is also right. All four angles must be right, making the parallelogram a rectangle by definition.

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Question

Given: Quadrilateral such that $m \angle A = 80 ^{\circ }$ and $m \angle B = 100^{\circ }$ .

True or false: It follows that Quadrilateral is a parallelogram.

Answer

$m \angle A + m \angle B = 80^{\circ } +100 ^{\circ } = 180^{\circ }$ , making $\angle A$ and $\angle B$ supplementary. By the Converse of the Same Side Interior Angles Theorem, , it does follow that $\overline{AD} || \overline{BC}$ . However, without knowing the measures of the other two angles, nothing further can be concluded about Quadrilateral . Below are a parallelogram and a trapezoid, both of which have these two angles of these measures.

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Question

Given: Parallelogram such that $\angle A \cong \angle B$ and $\angle C \cong \angle D$ .

True or false: It follows that Parallelogram is a rectangle.

Answer

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram can be proved to be supplementary - that is, their angle measures total $180^{\circ }$ . Specifically, $\angle A$ and $\angle B$ are a pair of supplementary angles. Since they are also congruent, it follows that both are right angles. For the same reason, $\angle C$ and $\angle D$ are also right angles. The parallelogram, having four right angles, is a rectangle by definition.

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Question

A rhombus has two interior angles with a measurement of degrees. What is the measurement of each of the other two interior angles?

Answer

The four interior angles in any rhombus must have a sum of degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of degrees.

Thus, if a rhombus has two interior angles of degrees, there must also be two angles that equal:

Check:

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