ACT Math : How to find an angle in a quadrilateral

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

Q6

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

Possible Answers:

\(\displaystyle 55^\circ\)

\(\displaystyle 70^\circ\)

\(\displaystyle 110^\circ\)

\(\displaystyle 145^\circ\)

\(\displaystyle 125^\circ\)

Correct answer:

\(\displaystyle 110^\circ\)

Explanation:

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

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

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

Therefore \(\displaystyle \angle ADB = 180-35-35 = 110\)

Example Question #431 : Plane Geometry

Q8

The rhombus above is bisected by two diagonals.

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

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

Possible Answers:

\(\displaystyle 15^\circ\)

\(\displaystyle 90^\circ\)

\(\displaystyle 50^\circ\)

\(\displaystyle 100^\circ\)

\(\displaystyle 80^\circ\)

Correct answer:

\(\displaystyle 80^\circ\)

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, \(\displaystyle \angle DAB,\angle ABC,\angle BCD,\) and \(\displaystyle \angle CDA\) are each bisected into two equal angles.

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

Thus,

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

\(\displaystyle 3x=45\)

\(\displaystyle x=15\) 

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

Since \(\displaystyle \angle EAB\) is only half of \(\displaystyle \angle DAB\),

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

\(\displaystyle \angle DAB=\angle BCD=80\)

Example Question #3 : How To Find An Angle In A Quadrilateral

Q10

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

 

Note: The figure may not be drawn to scale. 

Possible Answers:

\(\displaystyle 118^\circ\)

\(\displaystyle 124^\circ\)

\(\displaystyle 132^\circ\)

\(\displaystyle 121^\circ\)

\(\displaystyle 126^\circ\)

Correct answer:

\(\displaystyle 121^\circ\)

Explanation:

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

\(\displaystyle 15+2y=\frac{y+96}{2}\)

\(\displaystyle 30+4y=y+96\)

\(\displaystyle 3y=66\)

\(\displaystyle y=\frac{66}{3}=22\)

Therefore, \(\displaystyle \angle A = \angle C=59\)

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

\(\displaystyle \angle B = \frac{360-59-59}{2}=121\) 

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

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

Possible Answers:

\(\displaystyle \small 34^{o}\)

\(\displaystyle \small 20^{o}\)

\(\displaystyle \small 71^{o}\)

\(\displaystyle \small 30^o\)

\(\displaystyle \small 51^{o}\)

Correct answer:

\(\displaystyle \small 51^{o}\)

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:

\(\displaystyle 71+9x+5x+3x=360\)

When we combine like terms, we get the following:

\(\displaystyle 71+17x=360\)

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

\(\displaystyle 71-71+17x=360-71\)

\(\displaystyle 17x=289\)

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

\(\displaystyle \frac{17x}{17}=\frac{289}{17}\)

\(\displaystyle x=17\)

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: 

\(\displaystyle 71^{\circ}\) or \(\displaystyle 3x^{\circ}\)

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

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

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

\(\displaystyle 51^{\circ}\)

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