GRE Coordinate Geometry, Lines, Angles and Triangles

Coordinate geometry, lines, angles and triangles look like one connected topic on GRE Quant, since every one of them puts points, slopes or angles on a plane or inside a shape. ETS treats them as two separate content areas, though. The official GRE Math Review files coordinate geometry under Algebra, section 2.8, and files lines, angles and triangles under Geometry, sections 3.1 and 3.3.

This guide covers slope, distance and midpoint on the coordinate plane, equations of lines, angle relationships around an intersection and along parallel lines, and triangle properties including similarity and the Pythagorean theorem. Worked examples appear throughout, along with the setup traps costing points most often on test day. Practice both content areas with our drills at three difficulty levels.

Where These Topics Sit in GRE Quant Content Areas

GRE Quantitative Reasoning organizes content into four areas: arithmetic, algebra, geometry and data analysis. For the full breakdown of every area, read our GRE Quantitative Reasoning overview. ETS's official Math Review places coordinate geometry, including slope, distance and equations of lines, inside the Algebra part of the review, next to the equation and function skills covered in our algebra guide.

The same Math Review places lines and angles and triangles inside the Geometry part, next to circles and polygons. For circle and polygon formulas, read our guide to circles, quadrilaterals and polygons. This guide keeps the two content areas together, since coordinate geometry problems often use triangles drawn on the plane, and triangle problems sometimes plot a shape on x and y axes.

Slope, Distance and Midpoint on the Coordinate Plane

Three formulas answer most coordinate geometry questions: slope, distance and midpoint. Each formula converts two ordered pairs into a single number or a new point. All three read the same two points, so one setup error carries into every answer built on it.

Slope Formula and What It Tells You

ETS defines the slope m of a line through two points, (x1, y1) and (x2, y2) with x1 not equal to x2, as rise over run: m = (y2 - y1) / (x2 - x1). A positive slope rises left to right, a negative slope falls left to right, a slope of 0 marks a horizontal line, and an undefined slope, from division by zero, marks a vertical line. Take the points (1, 2) and (4, 8). The slope is (8 - 2) / (4 - 1) = 6 / 3 = 2.

The Distance Formula, Built From the Pythagorean Theorem

ETS's Math Review derives the distance between two coordinate points by building a right triangle from the horizontal and vertical legs between them, then applying the Pythagorean theorem, rather than presenting a standalone distance formula. For points (x1, y1) and (x2, y2), the horizontal leg measures x2 - x1 and the vertical leg measures y2 - y1. Square both legs, add them, and take the square root to find the distance: d = the square root of (x2 - x1) squared plus (y2 - y1) squared.

Take the points (1, 2) and (4, 6). The horizontal leg is 4 - 1 = 3 and the vertical leg is 6 - 2 = 4. The distance is the square root of (9 + 16), or the square root of 25, which equals 5.

Midpoint of a Segment

A midpoint splits a line segment into two equal halves. ETS's Math Review defines a midpoint geometrically, as the point bisecting a line segment, and never states a coordinate version. On the coordinate plane, average the x-coordinates and average the y-coordinates of the two endpoints: midpoint = ((x1 + x2) / 2, (y1 + y2) / 2). Take the points (2, 3) and (8, 7), giving a midpoint of ((2 + 8) / 2, (3 + 7) / 2) = (5, 5).

Equations of Lines on the Coordinate Plane

A line's equation lets you plot it point by point, or compare it against a second line without a graph. Two skills cover almost every GRE question here: reading slope-intercept form, and comparing slopes to test for parallel or perpendicular lines. Both start from the slope, so pull the slope out first.

Slope-Intercept Form

Slope-intercept form, y = mx + b, names the slope m and the y-intercept b directly, the point where the line crosses the y-axis. Take the equation y = 3x - 2. The slope is 3 and the line crosses the y-axis at -2. Set x = 0 to confirm the intercept: y = 3(0) - 2 = -2.

Parallel and Perpendicular Lines

ETS states two lines are parallel when their slopes are equal, and perpendicular when their slopes are negative reciprocals of each other. A negative reciprocal flips the fraction and switches the sign, so a slope of 2 pairs with -1/2. The table below lists the parallel and perpendicular slope for three sample lines.

Angle Relationships: Vertical, Complementary, Supplementary and Parallel Lines

Angle questions on GRE Geometry test relationships around a single point and along a pair of parallel lines cut by a third line. The rule set is short and fixed, so most of the work is reading the figure correctly. Mark each angle you solve directly on the figure, since one known angle usually settles the rest.

Vertical Angles and Angles Around a Point

ETS states opposite, or vertical, angles formed where two lines cross are congruent, and the four angles around the intersection sum to 360 degrees. Two lines cross and form four angles, one of them measuring 70 degrees. The angle opposite it also measures 70 degrees, since vertical angles are congruent. Each of the two remaining angles measures 110 degrees, since 360 - 70 - 70 = 220 splits evenly between them.

Complementary and Supplementary Angles

Two angles are complementary when they sum to 90 degrees, and supplementary when they sum to 180 degrees. A right angle splits into two complementary angles, and a straight angle splits into two supplementary angles. An angle of 35 degrees and an angle of 55 degrees are complementary, since 35 + 55 = 90. An angle of 110 degrees and an angle of 70 degrees are supplementary, since 110 + 70 = 180. ETS's Math Review never uses either word, so learn the underlying relationship and treat the two labels as prep shorthand.

Parallel Lines Cut by a Transversal

ETS covers a transversal crossing two parallel lines by noting the resulting angles take only two measures, x degrees and y degrees, where x + y = 180. Eight angles appear in the figure, and each one equals x or y. Every angle formed at the two intersections falls into one of two groups.

• Angles equal to the first measure, x degrees, sit in matching positions at both intersections.
• Angles equal to the second measure, y degrees, sit in the remaining positions, and x + y = 180.

If one angle measures 65 degrees, every angle in its group also measures 65 degrees. Every angle in the other group measures 180 - 65 = 115 degrees. One given angle therefore settles the whole figure.

Triangle Properties: Angle Sum, Similarity and the Pythagorean Theorem

Triangle questions on the GRE test a small, fixed set of rules, repeated across many shapes and setups. Four rules carry most of the work: the 180 degree angle sum, the special triangle types, similarity, and the Pythagorean theorem. Learn the condition attached to each rule, since the common error is applying a right-triangle rule to a triangle without a right angle.

Triangle Angle Sum and Polygon Angles

ETS states the interior angles of any triangle sum to 180 degrees, and this rule extends to every polygon. A polygon with n sides has interior angles summing to (n - 2) times 180 degrees, since a polygon splits into (n - 2) triangles. Take a triangle with angles of 50 degrees and 70 degrees. The third angle measures 180 - 50 - 70 = 60 degrees.

Special Triangle Types

ETS's Triangles section covers equilateral triangles, with all three angles at 60 degrees, isosceles triangles, with congruent base angles, and right triangles, including two special side ratios worth memorizing. The two ratios, 45-45-90 and 30-60-90, finish a right triangle from a single known side. The table below lists the defining rule for each type.

A 30-60-90 triangle has a shortest side of 5. The side opposite the 60 degree angle measures 5 times the square root of 3, and the hypotenuse measures 10, following the 1 to square-root-of-3 to 2 ratio. Match the shortest side to the 30 degree angle first, then scale the other two sides by the same factor.

Similar Triangles

ETS defines similar triangles as triangles sharing the same shape, with congruent corresponding angles and proportional corresponding sides. Every 30-60-90 right triangle is similar to every other 30-60-90 right triangle, regardless of size. Similarity gives you a proportion, so one matching pair of sides sets the scale factor for every remaining pair.

Triangle A has sides 3, 4 and 5. Triangle B has sides 6, 8 and 10, exactly double every side of triangle A. The two triangles are similar, with a scale factor of 2.

The Pythagorean Theorem

The Pythagorean theorem applies only to right triangles: a squared plus b squared equals c squared, where a and b are the two legs and c is the hypotenuse, the side opposite the right angle. ETS's Triangles section covers the Pythagorean theorem alongside the two special right-triangle ratios. Take a right triangle with legs of 6 and 8. The hypotenuse measures the square root of (36 + 64), or the square root of 100, which equals 10.

Common Traps on Coordinate Geometry and Triangle Questions

A small set of setup errors accounts for most missed points across both content areas. The same mistakes repeat on slope questions and triangle questions at every difficulty level. Each one produces a clean-looking wrong answer, so check the setup before you commit.

Sign Errors on Slope

Slope errors come from a handful of repeated setup mistakes. None of them break the arithmetic, so the wrong answer looks finished. Watch for these four patterns.

• Swapping the order of the two points in only one part of the slope formula, pairing (y2 - y1) with (x1 - x2) instead of (x2 - x1).
• Reading a falling line as a positive slope, or a rising line as a negative slope, from a graph.
• Mixing up the parallel rule, equal slopes, with the perpendicular rule, negative reciprocal slopes.
• Forgetting a vertical line has an undefined slope, not a slope of 0.

Misapplied Triangle Rules

Triangle errors come from applying a rule outside the shape it covers. A figure drawn to look like a right triangle or an isosceles triangle proves nothing without stated measurements or marks. Confirm the condition first, then apply the rule.

• Applying the Pythagorean theorem to a triangle without a right angle.
• Assuming two triangles are similar without confirming congruent angles or proportional sides.
• Mislabeling the hypotenuse in a right triangle drawn at an angle other than upright.
• Assuming a triangle is isosceles or equilateral without a stated or marked equal-side condition.

We offer three difficulty levels of drills for Coordinate Geometry and for Lines, Angles & Triangles on our GRE practice page, along with full-length practice tests and Quant section practice. Start at the level matching your last practice score, then move up. Every drill set is free with no signup required.

Sources

• ETS: GRE Math Review (PDF)