Geometry Points, Lines, Segments and Angles Lesson (Free PDF)

Geometry math lesson: Points, Lines, Segments and Angles – Maths PDF

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This basics lesson for Geometry covers points, lines, segments and angles (Common Core G-CO.A.1), with a free PDF to print.

A segment is the part of a line between two endpoints, and a ray starts at one point and goes on forever. A midpoint splits a segment into two equal parts, and an angle bisector splits an angle into two equal angles. Segment and angle addition give equations such as AB + BC = AC for unknown lengths and angles.

Geometry · Common Core State Standards: G-CO.A.1

Key facts in this basics lesson

Key facts

Point, line and plane are undefined terms; segments, rays and angles are defined from them.
Segment addition: if B is between A and C, then AB + BC = AC.
A midpoint at coordinate (a + b)/2 splits a segment into two congruent segments.
Angle addition: m∠ABD + m∠DBC = m∠ABC when ray BD is inside the angle.
A circle is all points in a plane at one fixed distance (the radius) from the center.

What are points, lines and planes?

Geometry starts from a few ideas that are described but not defined with other words. A point names a location and has no size. A line is a straight path that goes on forever in both directions. A plane is a flat surface that goes on forever. Every other definition in the course is built from these three.

Segment and ray

A line segment is the part of a line between two endpoints and has a length. A ray starts at one endpoint and goes on forever in one direction.

A line, a ray and a segment.

How do you find the length and midpoint of a segment?

On a number line, the length of the segment from \(a\) to \(b\) is \(|b – a|\). The midpoint is the point that splits a segment into two congruent parts, and its coordinate is the average \(\frac{a + b}{2}\).

If B is between A and C, then AB + BC = AC.

The Segment Addition Postulate says that when B is between A and C, \(AB + BC = AC\). Use it to write an equation when a length has a variable.

What is an angle and how do angles add?

An angle is formed by two rays with the same endpoint, called the vertex. Its measure is written \(m\angle ABC\) and is given in degrees. An angle bisector is a ray inside the angle that splits it into two angles with equal measures.

Angle addition: m∠ABD + m∠DBC = m∠ABC, so 35° + 48° = 83°.

Angle relationships

Two angles are complementary when their measures add to \(90^{\circ}\) and supplementary when they add to \(180^{\circ}\). Two lines are perpendicular when they meet at a \(90^{\circ}\) angle. Two lines in the same plane are parallel when they never meet.

How is a circle defined?

A circle is the set of all points in a plane that are the same distance from a fixed point called the center. That distance is the radius.

Every point of the circle is 5 units from the center O.

How do you solve a segment or angle problem with a variable?

Draw the figure and label every known length or angle measure.
Choose the postulate that fits: segment addition, angle addition, midpoint or bisector.
Write an equation and solve for the variable.
Substitute back to find each part, then check that the parts add to the whole.

Example: Point B is between A and C, \(AB = 3x + 1\), \(BC = 2x + 4\) and \(AC = 30\). Then \(5x + 5 = 30\), so \(x = 5\), \(AB = 16\) and \(BC = 14\). Check: \(16 + 14 = 30\).

Basics lesson: questions and answers

What is the difference between a line, a ray and a segment?

A line goes on forever in both directions, a ray has one endpoint and goes on forever in one direction, and a segment has two endpoints and a measurable length.

How do you find the midpoint of a segment?

Average the coordinates of the endpoints. On a number line from 4 to 18 the midpoint is (4 + 18)/2 = 11. In the plane, average the x-values and the y-values separately.

What does an angle bisector do?

It splits an angle into two angles with equal measures. If a 76 degree angle is bisected, each new angle measures 38 degrees.

What is the difference between complementary and supplementary angles?

Complementary angles add to 90 degrees and supplementary angles add to 180 degrees. A 37 degree angle has a 53 degree complement and a 143 degree supplement.

Why are point, line and plane called undefined terms?

Any definition would need other geometric words, which would need definitions in turn. So geometry describes these three ideas and uses them to define everything else precisely.

How to use this basics lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the basics lesson as a PDF with the button below and keep it with your notes.

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