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ACT Math : How to find an angle in a quadrilateral

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Example Question #1 : How To Find An Angle In A Quadrilateral

$\angle ADB$ bisects $\overline{EC}$ . If $\angle DAB=35$ then, in degrees, what is the value of $\angle ADB$ ?

Possible Answers:

$125^\circ$

$110^\circ$

$70^\circ$

$55^\circ$

$145^\circ$

Correct answer:

$110^\circ$

Explanation:

A rectangle has two sets of parallel sides with all angles equaling 90 degrees.

Since $\angle ADB$ bisects $\overline{EC}$ into two equal parts, this creates an isosceles triangle ABD .

Therefore $\angle DAB = \angle DBA$ . The sum of the angles in a triangle is 180 degrees.

Therefore $\angle ADB = 180-35-35 = 110$

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Example Question #2 : How To Find An Angle In A Quadrilateral

The rhombus above is bisected by two diagonals.

If $\angle ADE = x+35$ and $\angle EAB = 2x+10$ then, in degrees, what is the value of the $\angle BCD$ ?

Note: The shape above may not be drawn to scale.

Possible Answers:

$80^\circ$

$50^\circ$

$90^\circ$

$15^\circ$

$100^\circ$

Correct answer:

$80^\circ$

Explanation:

A rhombus is a quadrilateral with two sets of parallel sides as well as equal opposite angles. Since the lines drawn inside the rhombus are diagonals, $\angle DAB,\angle ABC,\angle BCD,$ and $\angle CDA$ are each bisected into two equal angles.

Therefore, $\angle ADE=\angle ABE$ , which creates a triangle in the upper right quadrant of the kite. The sum of angles in a triangle is 180 degreees.

Thus,

180 = (x+35)+(2x+10)+90=3x+135

3x=45

x=15

$\angle EAB = 2(15)+10=40$

Since $\angle EAB$ is only half of $\angle DAB$ ,

$\angle DAB=2(40)=80$

$\angle DAB=\angle BCD=80$

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Example Question #1 : How To Find An Angle In A Quadrilateral

Q10

If $\angle A = 15+2y$ and $\angle C=\frac{y+96}{2}$ , then, in degrees, what is the value of $\angle B$ ?

Note: The figure may not be drawn to scale.

Possible Answers:

$124^\circ$

$118^\circ$

$132^\circ$

$126^\circ$

$121^\circ$

Correct answer:

$121^\circ$

Explanation:

In a rhombus, opposite angles are equal to each other. Therefore we can set $\angle A$ and $\angle C$ equal to one another and solve for :

$15+2y=\frac{y+96}{2}$

30+4y=y+96

3y=66

$y=\frac{66}{3}=22$

Therefore, $\angle A = \angle C=59$

A rhombus, like any other quadrilateral, has a sum of angles of 360 degrees.

$\angle B = \frac{360-59-59}{2}=121$

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Example Question #4 : How To Find An Angle In A Quadrilateral

The interior angles of a quadrilateral are $71^{\circ}$ , $9x^{\circ}$ , $5x^{\circ}$ , and $3x^{\circ}$ . What is the measure of the smallest angle of the quadrilateral?

Possible Answers:

$\small 34^{o}$

$\small 71^{o}$

$\small 30^o$

$\small 51^{o}$

$\small 20^{o}$

Correct answer:

$\small 51^{o}$

Explanation:

In order to solve this problem we need the following key piece of knowledge: the interior angles of a quadrilateral add up to 360 degrees. Now, we can write the following equation:

71+9x+5x+3x=360

When we combine like terms, we get the following:

71+17x=360

We will need to subtract 71 from both sides of the equation:

71-71+17x=360-71

17x=289

Now, we will divide both sides of the equation by 17.

$\frac{17x}{17}=\frac{289}{17}$

x=17

We now have a value for the x-variable; however, the problem is not finished. The question asks for the measure of the smallest angle. We know that the smallest angle will be one of the following:

$71^{\circ}$ or $3x^{\circ}$

In order to find out, we will substitute 17 degrees for the x-variable.

$3(17^{\circ})=51^{\circ}$

Because 51 degrees is less than 71 degrees, the measure of the smallest angle is the following:

$51^{\circ}$

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ACT Math : How to find an angle in a quadrilateral

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find An Angle In A Quadrilateral

Example Question #2 : How To Find An Angle In A Quadrilateral

Example Question #1 : How To Find An Angle In A Quadrilateral

Example Question #4 : How To Find An Angle In A Quadrilateral

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