Angles

Angle and intersecting line questions only have a few core rules to memorize!

Opposite angles are always equal

When two lines intersect they create two pairs of equal angles, and the angles opposite from each other are always equal. The top left angle will equal the bottom right angle, and the top right angle will equal the bottom left angle.

Let’s try an example problem that involves this rule. Remember that a circle has 360 degrees, which also means that all the angles inside of a circle will add up to 360 degrees!

The circle drawn below has three lines that intersect at the center of the circle.

Quantity A: $x$
Quantity B: $y$

Feel free to take a moment to try to solve the question yourself, then keep reading to follow along!

The first step to this problem is to fill in all the opposite angles.

Since the sum of all the angles in a circle is 360, you can take these values and set up an equation to solve for $y$.

$$\begin{array}{rl}91+43+y+91+43+y& =360\\ 268+2y& =360\\ 2y& =92\\ y& =46\end{array}$$

With $y=46$ and $x=43$, Quantity B is greater.

Angles add up to equal greater angles

The second rule relates to the sum of angles within another angle. Essentially, the angles inside another angle will add up to the exterior angle. For example, in the first image in this chapter with the two intersecting lines, the sum of all four angles is 360 degrees, because all the angles would make a circle (imagine that the four angles are slices of a pie).

Now let’s imagine only half of the pie. What must angles $x$ and $y$ add up to below?

Any straight line has 180 degrees total, so in this example, $x+y=180$. Allow us to demonstrate further with this 80-degree angle below.

What do you think the sum of angles $x$ and $y$ is in this case?

What is the value of $x$ if $y$ was $40$?

Now let’s look at this angle ABC in the top right of the figure below.

What is the value of angle DBE?

What about angle DBA?

Let’s use the rule you just learned on this example problem.

What must be $x$ in terms of $z$ if $x$ is twice the value of $y$? Line $AB$ is a straight line.

A. $1/2z$
B. $z$
C. $2z$
D. $3z$
E. $4z$

Can you figure out the answer?

Because $AB$ is a straight line, we can make the equations $180=z+x$ and $180=y+2z$. And since they’re both equal to $180$, we can combine them into a single equation, and solve.

$$\begin{array}{rl}z+x& =y+2z\\ x& =y+z\end{array}$$

Now we have $x$ in terms of $y$ and $z$, but the question asks for the value of $x$ in terms of only $z$. The question also tells us that $x$ is twice the value of $y$, i.e. $x=2y$, or put another way, $y=x/2$. So we can plug this in:

$$\begin{array}{rl}x& =y+z\\ x& =x/2+z\\ x/2& =z\\ x& =2z\end{array}$$

So finally, in terms of $z$, $x$ can be expressed as $2z$.

Now let’s solidify what we’ve learned by doing a practice question together. Follow along with this video: