04. Interior and Exterior Angles

Question 1

ABCD is a quadrilateral.

Find the value of the missing angle labelled x.

Select the correct answer from the list below:

A: 74^\circ

B: 42^\circ

C: 62^\circ

D: 54^\circ

CORRECT ANSWER:  C: 62^\circ

WORKED SOLUTION:

The polygon we are looking at is a quadrilateral (four sides), so we know that the angles inside have to add up to 180\times(4-2)=360^\circ. However, we don’t know angle ABC, so we need to find that first.

Angle ABC

We know that angles on a straight line add up to 180^\circ, so we can say:

126+y=180
y=180-126
y=54

Now, all we need to do is use the fact that the angles inside add up to 360^\circ.

x+138^\circ +54^\circ +106^\circ =360^\circ
x+298^\circ =360^\circ
x =360^\circ-298^\circ
x =62^\circ

Level 4

Question 2

ABC is a triangle.

Find the value of the missing angle labelled x.

Select the correct answer from the list below:

A: 45^\circ

B: 65^\circ

C: 135^\circ

D: 90^\circ

CORRECT ANSWER: A: 45^\circ

WORKED SOLUTION:

There are three important pieces of information we need to consider:

1. The lines on the base and height mean that this triangle is an isosceles, so the two sides and angles must be the same.
2. The little box in the bottom left corner means that it is a right-angle (90^\circ).
3. The polygon is a triangle, so the angles will add up to 180^\circ.

Considering the first and second points, we can label the angle that is unlabelled and write in the 90^\circ.

And now, we can use the third point to find the value of x.

90^\circ +x+x=180^\circ
90^\circ +2x=180^\circ
2x=180^\circ-90^\circ
2x=90^\circ
x=90^\circ\div2
x=45^\circ

Level 4

Question 3

ABCDE is a pentagon.

Find the value of x.

Select the correct answer from the list below:

A: 34^\circ

B: 56^\circ

C: 27^\circ

D: 35^\circ

CORRECT ANSWER:A: 34^\circ

WORKED SOLUTION:

Because a pentagon has five sides, we need to put 5 into our formula for finding out how much angles in a polygon add up to.

So, we can say that all of the angles in our pentagon must add up to 540^\circ

81^\circ+111^\circ+110^\circ+2x+x+4x=540^\circ 302^\circ+7x=540^\circ

And now, all we need to do is rearrange to find x.

7x=540^\circ -302^\circ 7x=238^\circ x=238^\circ \div7 x=34^\circ

Level 4

Question 4

ABCDEF is a hexagon. The lines AB and ED are parallel, and so are AF and CD. Find the value of x

Select the correct answer from the list below:

A: 193^\circ

B: 201^\circ

C: 284^\circ

D: 237^\circ

CORRECT ANSWER:D: 237^\circ

WORKED SOLUTION:

Because a hexagon has 6 sides, we need to put 6 into our formula for finding out how much angles in a polygon add up to.

So, we can say that all of the angles in our hexagon must add up to 720^\circ. But, for this to help us, we need to figure out the missing angle, CDE. In fact, because of our parallel lines, ADE is the same as BAF. It can be helpful to picture it as a parallelogram.

And because opposite angles in a parallelogram are equal, we have that CDE = BAF.

And now, we can add all the angles in the hexagon and put them equal to 720^\circ.

If we collect the like terms this gives us

And now, we can rearrange to find z

9z=720^\circ-441^\circ
9z=279^\circ
z=279^\circ\div9
z=31^\circ

We can now substitute this value of z into our diagram to help us find x.

Now, because angles around a point add up to 360^\circ, we can say that.

Which we can rearrange to find x.

x =360^\circ -123^\circ
x =237^\circ

Level 4

Question 5

Below is part of a 15-sided polygon. Find the value of the angle x.

Select the correct answer from the list below:

A: 217^\circ

B: 232^\circ

C: 204^\circ

D: 195^\circ

CORRECT ANSWER:  C: 204^\circ

WORKED SOLUTION:

To start, we need to find how much the interior angles add up to, which we can do with our formula.

180^\circ\times(15-2)=180^\circ\times13=2340^\circ 15 \text{ Angles }=2340^\circ

But we don’t want 15, we only want 1. So, let’s divide this by 15.

And finally, because we know angles around a point add up to 360^\circ, we can say:

Which we can then rearrange to find x

x =360^\circ -156^\circ x =204^\circ

Level 4