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. Angles directly opposite each other are always equal:

• The top-left angle equals the bottom-right angle.
• The top-right angle equals the bottom-left angle.

Let’s try an example that uses this rule. Remember: a full circle measures 360 , so all angles around a point (like the center of a circle) add up to 360 .

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

Quantity A: $x$
Quantity B: $y$

Take a moment to try the question yourself, then read on to see one clear way to solve it.

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

Since the sum of all angles around the center is 360 , you can use the labeled values to set up an equation and 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 is about angle sums. If an angle is split into smaller angles, the measures of the smaller angles add up to the measure of the larger (outer) angle.

For example, in the first image in this chapter (two intersecting lines), all four angles together make a full circle, so their sum is 360 .

Now imagine only half of that “pie.” What must angles $x$ and $y$ add up to below?

Any straight line forms a 180 angle, so in this example, $x+y=180$.

Now look at the 80 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 look at angle ABC in the top right of the figure below.

What is the value of angle DBE?

What about angle DBA?

Let’s use this rule in an 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, each pair of adjacent angles on that line must add to $180$.

• From the top, $z+x=180$.
• From the bottom, $y+2z=180$.

Since both expressions equal $180$, set them equal to each other and solve:

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

Now you have $x$ in terms of $y$ and $z$, but the question asks for $x$ in terms of only $z$. You’re also told that $x$ is twice $y$, meaning $x=2y$, or equivalently $y=x/2$. Substitute $y=x/2$ into the equation:

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

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

Now let’s solidify what you’ve learned with a practice question. Follow along with this video: