8.1 Lines, Circles, Triangles, Quadrilaterals & Polygons

Lines, Circles, Triangles, Quadrilaterals & Polygons

Quick Answer: GRE geometry rewards property recall over measurement: angle pairs from parallel lines cut by a transversal, triangle angle-sum and inequality rules, circle tangent/arc/inscribed-angle relationships, and the (n − 2) × 180° polygon angle formula cover the bulk of items. Because figures are not drawn to scale, only stated measures and proven relationships count.

The ETS Math Review explicitly states: geometric figures are NOT drawn to scale unless noted. Lengths that look equal may differ; angles that look right may not be. Coordinate axes and graphical data ARE to scale. This single convention changes how you read every geometry question — extract all measures from text and labels, then reason, never eyeball.

Lines, Segments, Rays, and Angles

A line extends infinitely in both directions. A line segment has two endpoints; its length is the distance between them. A ray has one endpoint and extends infinitely in one direction. Two lines in a plane are parallel (never meet) or intersecting; intersecting lines form four angles whose adjacent pairs are supplementary (sum to 180°) and whose vertical (opposite) angles are equal.

Angle types by measure:

Parallel Lines Cut by a Transversal

When a transversal crosses two parallel lines, eight angles form four equal-or-supplementary pairs. Memorize these names — they appear in GRE stems:

• Corresponding angles: equal (same position at each crossing).
• Alternate interior angles: equal (on opposite sides of the transversal, between the parallels).
• Alternate exterior angles: equal (on opposite sides, outside the parallels).
• Same-side interior angles: supplementary (sum to 180°).

Worked Example — Find the Missing Angle

Two parallel lines are cut by a transversal. One interior angle is labeled 65°. Find the angle that is alternate interior to it and the angle that is same-side interior to it.

• Alternate interior angle = 65° (equal).
• Same-side interior angle = 180° − 65° = 115° (supplementary).

Circles

A circle is the set of points equidistant from a center. Key vocabulary:

• Radius (r): distance from center to any point on the circle.
• Diameter (d = 2r): longest chord, through the center.
• Chord: segment with both endpoints on the circle.
• Tangent: a line that touches the circle at exactly one point; perpendicular to the radius at that point.
• Arc: a portion of the circumference; its measure equals the central angle that intercepts it.
• Central angle: vertex at the center.
• Inscribed angle: vertex on the circle, sides are chords; measure is half the intercepted arc.

Formulas: circumference C = 2πr = πd; area A = πr².

Worked Example — Inscribed vs. Central Angle

A central angle of 80° intercepts arc AB. An inscribed angle intercepts the same arc AB. The inscribed angle is 80° / 2 = 40°. This 2-to-1 ratio is a frequent QC shortcut.

Triangles

Every triangle satisfies three core rules:

1. Angle sum: interior angles sum to 180°.
2. Exterior angle: an exterior angle equals the sum of the two remote interior angles.
3. Triangle inequality: any two sides sum to more than the third; equivalently, the third side is between |a − b| and a + b.

Triangle area: A = ½ × base × height, where height is perpendicular to the chosen base.

Triangle types by sides: equilateral (all sides equal, all angles 60°), isosceles (two equal sides, two equal angles opposite them), scalene (no equal sides). By angles: acute (all angles < 90°), right (one 90° angle), obtuse (one angle > 90°).

Worked Example — Triangle Inequality

Two sides of a triangle are 7 and 10. What is the range of the third side x?

• Lower bound: |10 − 7| = 3.
• Upper bound: 10 + 7 = 17.
• Range: 3 < x < 17. (Strict inequalities — equality would produce a degenerate triangle.)

Quadrilaterals and Their Areas

A square is both a rectangle and a rhombus; a rectangle and a rhombus are both parallelograms. The GRE sometimes asks which category contains another — keep the hierarchy crisp.

Polygons

A polygon with n sides has interior angle sum (n − 2) × 180°. For a regular polygon (equal sides and angles), each interior angle measures (n − 2) × 180° / n, and each exterior angle measures 360° / n.

Worked Example — Find the Missing Angle in a Pentagon

A pentagon has four interior angles measuring 100°, 110°, 120°, and 130°. Find the fifth.

• Sum of all five = (5 − 2) × 180° = 540°.
• Sum of the four known = 460°.
• Fifth angle = 540° − 460° = 80°.

Why This Matters for the GRE

Geometry items on the GRE are rarely about computation alone — they test whether you can extract the right property from a deliberately misleading figure. When a question shows two triangles that look congruent but labels only one pair of equal sides, the figure is bait; only the stated equality is real. Build the habit of listing all labeled measures before reasoning, and treat every geometric figure as a schematic, not a picture.