Lines and angle relationships

01 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

02 / UNDERSTAND & APPLY

Read a geometric figure as information

An angle measures the amount of turning between two rays with a common endpoint. In angle ABC, B is the vertex. Extending or shortening the drawn rays changes neither their direction nor the angle. Length and angle are different quantities: a long-looking ray does not imply a larger angle.

For ordinary GRE geometry sketches, use stated measurements, right-angle marks, equality marks, and specified relationships. Do not measure the drawing or assume that apparently horizontal lines are parallel. Straightness, the order of points along a line, and the relative arrangement shown do carry information. Coordinate axes, number lines and data graphs follow a different convention: they are drawn to scale.

Before calculating, mark every given fact and identify the exact angle requested. A diagram that looks symmetric may permit asymmetric cases. Redrawing an allowed configuration is often more informative than trying to make the original picture more precise.

03 / UNDERSTAND & APPLY

Use degree benchmarks and partitions

A full turn is 360°, a straight angle is 180°, and a right angle is 90°. An acute angle lies strictly between 0° and 90°; an obtuse angle lies strictly between 90° and 180°. A reflex angle is greater than 180° and less than 360°. When a question names the smaller angle between two rays, exclude the reflex alternative.

Adjacent nonoverlapping angles add when their interiors partition a larger angle. Two rays inside a right angle may divide it into several parts; the sum is 90°, regardless of the apparent spacing. Angles around a point sum to 360° only if you include exactly one full turn, without overlapping or skipping a region.

04 / UNDERSTAND & APPLY

Name the relationship before solving

Complementary angles sum to 90°; supplementary angles sum to 180°. Neither relationship requires the angles to be adjacent. A linear pair is a special case: two adjacent angles whose noncommon sides are opposite rays. That physical straight-line condition guarantees supplementary measures.

Being next to one another does not by itself make two angles supplementary. If three small angles split a straight angle, all three sum to 180°, not every pair. Likewise, if one angle is described as twice another, translate the ratio before choosing a sum.

05 / UNDERSTAND & APPLY

Separate vertical angles from adjacent angles

Two distinct intersecting straight lines form two pairs of vertical angles. Vertical angles are opposite at the vertex and have equal measures. Adjacent angles formed by the two lines are a linear pair and are supplementary. These facts hold without any parallel-line assumption.

If one angle is t°, the four measures alternate t°, 180° − t°, t°, 180° − t°. They are all equal only when t = 90. In a drawing with more than two lines, identify the particular opposite rays before declaring angles vertical; sharing a vertex alone is insufficient.

06 / UNDERSTAND & APPLY

Carry an angle through parallel lines

A transversal intersects two other lines at different points. If those two lines are parallel, corresponding angles are equal: they occupy the same relative position at their intersections. Alternate interior angles, which are between the parallels on opposite sides of the transversal, are also equal. Same-side interior angles are supplementary.

In the labeled figure, a = e by corresponding positions and d = f by alternate interior positions. Angles c and f are same-side interior angles, so c + f = 180°. Vertical pairs at each intersection supply additional equalities. Trace positions instead of relying on remembered letter shapes such as “Z” or “F,” which can become confusing when a sketch is rotated.

The condition “parallel” is indispensable. Without it, you can still use vertical and linear-pair facts at each intersection, but you cannot freely transport a measure to the other intersection.

07 / UNDERSTAND & APPLY

Use perpendicularity without inventing it

Perpendicular lines intersect at right angles. Once one angle between two straight lines is known to be 90°, the vertical and supplementary relationships force all four to be 90°. A square corner mark or an explicit perpendicular statement supplies this fact; a nearly square-looking corner does not.

In a plane, two distinct lines perpendicular to the same line are parallel. A line perpendicular to one of two parallel lines is perpendicular to the other. These relationships let you connect facts across a figure, but require the stated common line and the planar setting.

08 / UNDERSTAND & APPLY

Connect angle equations with valid ranges

Solve only after choosing the geometric equation. A variable is not necessarily itself an angle: if an angle is 5x − 20 degrees, it is that expression that must fit the geometry. In an ordinary nondegenerate intersection, each smaller angle is strictly between 0° and 180°. A negative measure or a measure outside this interval signals a wrong equation or an inadmissible root.

Ratios are often faster than a general equation. If a supplementary pair is in the ratio 4:5, there are nine equal shares in 180°, so each share is 20° and the angles are 80° and 100°. An angle bisector creates two equal angles; it does not necessarily bisect a nearby side or distance.

• Translate “more than” as an addition to a measure, not to the final sum.
• Keep degree units on angle measures and distinguish the parameter from the answer.
• If only a sum is fixed, individual addends usually remain free.

09 / UNDERSTAND & APPLY

Recognize insufficient angle information

A QC relationship must hold in every configuration allowed by the givens. Two complementary positive angles can be equal or unequal. Knowing their sum does not identify which is larger. To demonstrate insufficient information, exhibit allowed measures that change the comparison; do not merely state that no diagram is supplied.

Sometimes the actual measures are unknown but the comparison is fixed. If positive x and y satisfy x + y = 180 and x is acute, then y is obtuse and y > x. A useful bound can replace a full calculation.

10 / UNDERSTAND & APPLY

Practice: angle structure and exact directions

For QC use A = Quantity A greater, B = Quantity B greater, C = equal, D = not determined. For all other items follow the selection or numeric-entry instruction stated in the question.

11 / UNDERSTAND & APPLY

Practice: conditions and counterexamples

Check that each proposed relationship has its required condition. A correct arithmetic operation on an unjustified equation still produces an invalid solution.