1Point
A point is an exact location in space. It has no size, no length and no width, only a position. We draw it as a dot.
Name a point with one capital letter: point A.
In real life: A city marked on a map.
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Geometry is the part of math that studies points, lines, angles, surfaces and solids. This lesson explains each basic idea with a definition, a picture, the way to write it, and an example from real life. Check yourself with the short quiz above.
A point is an exact location in space. It has no size, no length and no width, only a position. We draw it as a dot.
Name a point with one capital letter: point A.
In real life: A city marked on a map.
A line is a straight path of points that goes on forever in both directions. It has length but no width or thickness. The arrows show that it never ends.
Name a line with any two of its points: line AF or line FA, written AF. A line can also be named with one lowercase letter, such as line m.
In real life: A long straight road, imagined going on forever both ways.
A line segment is the part of a line between two endpoints. Unlike a line, it has a definite length that you can measure.
Name a segment by its endpoints: segment AF, written AF.
In real life: The stretch of a road between two intersections, or the edge of a ruler.
A ray starts at one endpoint and goes on forever in one direction.
Name a ray with its endpoint first, then another point on it: ray AF, written AF. Ray FA would start at F and point the other way.
In real life: A beam of light from a flashlight or a laser.
A plane is a flat surface that goes on forever in all directions. It has length and width but no thickness. We draw part of it as a slanted box.
Name a plane with three points on it that are not on one line, such as plane ABC, or with one capital letter, such as plane P.
In real life: A tabletop or a sheet of paper, imagined stretching forever.
Points are collinear if one straight line passes through all of them. Points are coplanar if they all lie in the same plane.
Points A, B and C are collinear. Point D is not on that line, so A, B and D are not collinear. All four points are coplanar: they lie on this page.
In real life: Trees planted in one straight row are collinear.
Parallel lines lie in the same plane and never meet, however far they are extended. They are always the same distance apart.
Lines m and n are parallel, written m ∥ n. Matching arrow marks on the lines show they are parallel.
In real life: Railroad tracks, or the lines on notebook paper.
Intersecting lines cross each other. Two different lines can cross at only one point.
Lines p and q intersect at point X.
In real life: Two roads crossing at an intersection.
Perpendicular lines intersect at a right angle (90°). The small square at the crossing is the sign for a right angle.
Lines r and s are perpendicular, written r ⊥ s.
In real life: The edges at the corner of a sheet of paper, or a plus sign.
The midpoint of a segment is the point that divides it into two equal parts. Segments with the same length are congruent. Matching tick marks show congruent segments.
M is the midpoint of AB, so AM ≅ MB (both are 4 cm).
In real life: The middle of a seesaw, where it balances.
An angle is formed by two rays that share the same endpoint. The shared endpoint is the vertex, and the rays are the sides. Angles are measured in degrees (°).
This is angle FAC, written ∠FAC or ∠CAF, made by rays AF and AC. The vertex A always goes in the middle of the name. Its measure is m∠FAC = 50°.
In real life: The hands of a clock, or a slice of pizza.
A vertex is a point where two or more rays, lines, segments or edges meet. The plural is vertices.
The blue points below are vertices: the corner of an angle, the crossing of two lines, a corner of a triangle and a corner of a cube.
In real life: The corner of a room, where two walls and the floor meet.
Angles are named by their size. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is between 90° and 180°. A straight angle is exactly 180°. A reflex angle is more than 180°.
A right angle is marked with a small square instead of an arc.
In real life: A slice of pizza (acute), the corner of a book (right), an open laptop (obtuse).
Two angles are complementary if their measures add up to 90°, and supplementary if they add up to 180°. When two lines cross, the angles opposite each other are vertical angles, and vertical angles are always equal.
35° + 55° = 90°, so they are complementary. 60° + 120° = 180°, so they are supplementary.
In real life: Scissors: when the blades open, the opposite angles stay equal.
A polygon is a closed flat shape made of straight sides (segments) that meet only at their endpoints. Each corner is a vertex. A polygon is named by its number of sides.
3 sides: triangle, 4: quadrilateral, 5: pentagon, 6: hexagon, 8: octagon, 10: decagon. If all sides and all angles are equal, the polygon is regular.
In real life: A stop sign is an octagon; a honeycomb cell is a hexagon.
A circle is the set of all points in a plane that are the same distance from one point, the center. A radius goes from the center to the circle. A diameter goes across the circle through the center, so it is twice the radius: d = 2r. A chord is any segment with both endpoints on the circle.
The circle is named by its center: circle O. The diameter is the longest chord.
In real life: A clock face, a wheel, a coin.
A solid (3D shape) has length, width and height. Flat surfaces are faces, the segments where two faces meet are edges, and the corners are vertices.
A cube has 6 faces, 12 edges and 8 vertices. Cylinders, cones and spheres have curved surfaces.
In real life: A die (cube), a can (cylinder), an ice-cream cone (cone), a ball (sphere).
These ideas are the building blocks for every shape, formula and proof in geometry. Practice by finding points, lines, angles and shapes around you, and by drawing and naming them yourself.
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