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Mountain with a flag at the peak
Textbook
1. Welcome
2. Vocabulary approach
3. Quantitative reasoning
3.1 Quant intro
3.2 Arithmetic & algebra
3.3 Statistics and data interpretation
3.4 Geometry
3.4.1 Angles
3.4.2 Triangle basics
3.4.3 Sum of interior angles
3.4.4 Pythagorean theorem
3.4.5 Right triangles (45-45-90)
3.4.6 Right triangles (30-60-90)
3.4.7 Triangle inequality theorem
3.4.8 Coordinate plane
3.4.9 Equation for a line
3.4.10 Graphing inequalities
3.4.11 Graphing parabolas
3.4.12 Graphing circles
3.4.13 Parallel and perpendicular lines
3.4.14 Quadrilaterals
3.4.15 Circles
3.4.16 3D shapes
3.4.17 Polygons
3.4.18 Regular polygons
3.4.19 Shaded region problems
3.5 Strategies
4. Verbal reasoning
5. Analytical writing
6. Wrapping up
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3.4.1 Angles
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3. Quantitative reasoning
3.4. Geometry

Angles

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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.

equal opposite angles

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

Circle with interior angles

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.

Circle with interior 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.

91+43+y+91+43+y268+2y2yy​=360=360=92=46​

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?

complementary angles

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.

bisected angle


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

(spoiler)

If you guessed 80, then you would be correct!

What is the value of x if y was 40?

(spoiler)

Since the angles need to add up to 80, if y is 40, that means x is also 40.

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

Equal opposite angles with a measured angle


What is the value of angle DBE?

(spoiler)

DBE is equal to 80 because it is opposite ABC, and opposite angles are equal.

What about angle DBA?

(spoiler)

DBA is equal to 100. Together, ABC and DBA make a straight line, so the sum of their angles must equal 180.

If you have sharp eyes, you might notice that the angle above is not actually drawn as 80° in the figure.

Keep in mind that images, figures, and charts on the GRE will not necessarily be drawn to scale!

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

Line showing sum of angles


Can you figure out the answer?

(spoiler)

Answer: C. 2z

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.

z+xx​=y+2z=y+z​

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:

xxx/2x​=y+z=x/2+z=z=2z​

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:

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