GRE Quantitative: Lines and Angles

Lines and angles constitute the foundation of GRE geometry; though you might have learned your triangles and squares first, all polygons are essentially made up lines and angles. Here are the basics of lines and angles so you can nail those basic geometry questions.

Angles

An angle is formed by the union of two lines that share an endpoint (the vertex of an angle). The angle measurement corresponds to how far you have to rotate one of the lines to reach the other line.

Angles are measured in degrees, symbolized by the symbol º. No, that’s not an exponent of 0, but it looks pretty close. A complete rotation has 360 degrees, so it makes sense that a circle has 360 degrees, and the four angles produced by two intersecting lines (seen below) add up to 360 degrees.

While a full revolution is 360 degrees, a half revolution (a.k.a. a straight angle) has 180 degrees, and a quarter revolution (a.k.a. right angle) has 90 degrees.

Obtuse and Acute

You may remember that “acute” angles are less than 90 degrees while obtuse angles are more than 90 degrees but less than 180.

Complementary and Supplementary

The terms complementary and supplementary refer to special pairs of angles. Complementary angles add up to 90 degrees ad supplementary angles add up to 180 degrees.

Vertical Angles

When two lines intersect, we have two pairs of equal angles that are opposite each other.

In the diagram, angles 1 and 3 are equal and angles 4 and 2 are equal.

Parallel Lines

Lines that never intersect are called parallel lines. You may see this symbolized on the test as | |. Think of train tracks as parallel lines–they always run along each other and never converge.

Traversals

You will almost always run into at least one problem that presents two parallel lines intersected by a third straight line known as a traversal. When this happens, eight angles are formed with special relationships to each other. Essentially, you can figure out all eight angles when given only one angle.

In the diagram above, angles A, D, E, and H are equal to each other while angles B, C, F and G and equal to each other. The sum of any two adjacent angles, like A and B or F and H, is always 180 degrees (since they are supplementary). For example, if angle A was 110 degrees, and I asked you to find the rest of the angles, you would immediately know that D, E, and H are 110 degrees while B, C, F, and G each has 70 degrees (180 – 110= 70).

There are special names for these related angles in the diagram. In the diagram, angle pairs like A and H are alternate exterior angles, angle pairs like C and F are alternate interior angles, and angle pairs like A and E are corresponding angles.

Perpendicular Lines

Two intersecting lines that form 90 degrees (a.k.a. a right angle) are called perpendicular lines. Simply put, when two lines form a cross or a “T,” they are perpendicular.

Angles and lines are used in diagrams throughout the test. Knowing these basics will help you immensely with even the most complicated geometric diagrams.

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  • Sundizzz

    Thanks. This SIMPLE guide has helped so much. I didn’t want to learn too much about this, just needed to find a site with the basics to help me on this part of the test. Thanks so much. Is there something similar to this for triangles? Example, when there is a triangle PQR and there is a line that intersects from P to a point on QR.