Geometry - How to find an angle in a parallelogram

Rhombus_with_missing_angles_

Using the above rhombus, find the measurement of angle .

$110^\circ$

$70^\circ$

$55^\circ$

$60^\circ$

Explanation

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, $\measuredangle C=\measuredangle A$

Custom_vt_rhomb_3

Using the above rhombus, find the measurement of angle

$133^\circ$

$313^\circ$

$47^\circ$

$113^\circ$

Explanation

Thus, the solution is:

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

Custom_vt_rhomb_3

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

$94^\circ$

$47^\circ$

$90^\circ$

$133^\circ$

Explanation

Thus, the solution is:

$\measuredangle H=47^\circ$

$\measuredangle F=47^\circ$

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

Vt_rhomb_4

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

$256^\circ$

$128^\circ$

$104^\circ$

$65^\circ$

Explanation

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,

$128+128=256^\circ$

Vt_parallelogram_5

Using the parallelogram above, find the measurement of angle

$81^\circ$

$162^\circ$

$91^\circ$

$80^\circ$

Explanation

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:

$180-99=81^\circ$

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

True or false: Quadrilateral is a parallelogram.

False

True

Explanation

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

Pentagon a

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.

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

True or false: Parallelogram must be a rectangle.

True

False

Explanation

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.

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.

False

True

Explanation

$m \angle A + m \angle B = 80^{\circ } +100 ^{\circ } = 180^{\circ }$ , 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.

Parallelograms

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.

True

False

Explanation

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.