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Geometry Unit 1: Points, lines, angles & coordinates

Geometry starts with the vocabulary of points, lines and angles, plus two coordinate formulas you'll use all year.

Covers: Angle pairs (complementary, supplementary, vertical), midpoint and distance, angle bisectors

Study guide

Geometry starts with undefined terms: points, lines and planes. A segment is part of a line with two endpoints, a ray has one endpoint, and an angle is formed by two rays with a common vertex. Angles are classified as acute (under 90°), right (90°), obtuse (between 90° and 180°) or straight (180°).

Angle pairs have fixed relationships. Complementary angles add to 90°, supplementary angles add to 180°, and a linear pair (adjacent angles on a line) is supplementary. Vertical angles, the opposite angles formed when two lines intersect, are always equal. Many problems come down to writing an equation from one of these relationships and solving for x.

An angle bisector splits an angle into two equal angles, and a segment bisector splits a segment into two equal parts at its midpoint.

On the coordinate plane, the midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2): the average of the coordinates. The distance between them is √((x₂ − x₁)² + (y₂ − y₁)²), which is the Pythagorean theorem applied to the horizontal and vertical changes. These two formulas are the basis for coordinate proofs later in the course.

To find the point P that divides the directed segment from A(x₁, y₁) to B(x₂, y₂) in the ratio a:b, compute the fraction k = a/(a + b) and move that fraction of the way from A: P = (x₁ + k(x₂ − x₁), y₁ + k(y₂ − y₁)). The order matters, since the ratio is measured starting from A. The slope of a segment is (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes and perpendicular lines have slopes whose product is −1. To write the perpendicular bisector of a segment, find its midpoint, take the negative reciprocal of its slope, and use point-slope form y − y₁ = m(x − x₁).

Every point on the perpendicular bisector of a segment is the same distance from both endpoints. Conversely, any point that is equidistant from the endpoints lies on the perpendicular bisector. Basic constructions include copying a segment or angle, bisecting an angle, constructing a perpendicular bisector and drawing a line parallel to a given line. Each one relies on arcs with equal radii. To inscribe a regular hexagon or equilateral triangle in a circle, mark off arcs equal to the radius around the circle (six marks for a hexagon, every other mark for a triangle).

Key ideas

• Complementary angles sum to 90°; supplementary to 180°.
• Vertical angles are equal.
• A bisector splits an angle or segment into two equal parts.
• Midpoint averages the coordinates.
• Distance comes from the Pythagorean theorem.

Formulas and rules

Common mistakes

• Mixing up complementary and supplementary.
• Subtracting coordinates in the midpoint formula.

Words to know

Complementary angles

Two angles that add to 90°.

Supplementary angles

Two angles that add to 180°.

Vertical angles

Opposite angles formed by two intersecting lines; always equal.

Midpoint formula

((x₁ + x₂)/2, (y₁ + y₂)/2)

Distance formula

√((x₂ − x₁)² + (y₂ − y₁)²)

Point, line, plane

The undefined terms of geometry.

Segment

Part of a line with two endpoints.

Ray

Part of a line with one endpoint, extending forever in one direction.

Acute / right / obtuse angle

Less than 90°; exactly 90°; between 90° and 180°.

Linear pair

Adjacent angles forming a straight line; they're supplementary.

Angle bisector

A ray that splits an angle into two equal angles.

Segment bisector

A line, ray or segment through a segment's midpoint.

Sample practice questions

33 more questions like these, from level 1 to 5, are in the practice set.