Give precise definitions of segment, ray, angle, circle, perpendicular lines and parallel lines from the undefined notions of point, line, distance and arc length

Learning goal: State precise definitions of line segment, ray, angle, circle, perpendicular lines and parallel lines, built only from the undefined notions of point, line, distance along a line and arc length, and use each definition to decide whether a given object qualifies.

Why some notions are left undefined. Every definition uses words, and those words need definitions too. If the chain never stopped, nothing would ever be defined. Geometry breaks the chain deliberately: a small number of notions are accepted as undefined, described by examples and pictures rather than by a sentence, and everything else is defined in terms of them. The undefined notions in this course are point (a location with no size), line (a straight path with no thickness that extends without end in two directions), distance along a line (a non-negative number attached to any two points, written AB, which is 0 only when the points coincide) and arc length (the distance traveled along a circle between two of its points). A precise definition is one that uses only these notions and previously defined terms, and that includes every object it should and excludes every object it should not. Testing a definition means trying to build something that satisfies the words but is not the intended object.

Line segment. The line segment AB, for two distinct points A and B, is the set consisting of A, B and every point P of line AB for which AP + PB = AB. The condition AP + PB = AB is the precise way of saying "P lies between A and B" using only distance. A point Q on line AB beyond B has AQ = AB + BQ, which is larger than AB, so AQ + QB is larger than AB and Q is excluded. The length of segment AB is the distance AB. A segment therefore has two endpoints and a definite length, unlike a line, which has neither.

Ray. The ray AB, with endpoint A, is the set of points P on line AB for which A is not between P and B. In distance language, ray AB contains every point P of line AB with PA + AB ≠ PB, together with A itself. It starts at A, passes through B and continues without end in that one direction. Ray AB and ray BA are different objects: they share segment AB and point in opposite directions. A common student error is to treat rays as having two endpoints; a ray has exactly one.

Angle. An angle is the figure formed by two rays that share an endpoint; the shared endpoint is the vertex and the rays are the sides. The measure of an angle is defined through arc length: draw any circle centered at the vertex with radius r, and let s be the arc length cut off by the two sides on the shorter arc. The measure in degrees is 360 × s ÷ (2πr). This is a precise definition because it does not depend on which circle you choose: doubling the radius doubles both the arc length and the circumference, and the fraction s ÷ (2πr) is unchanged. A straight angle has measure 180° because its arc is exactly half the circle, and a right angle has measure 90° because its arc is a quarter of the circle.

Circle. A circle with center O and radius r (a positive number) is the set of all points P in the plane whose distance from O equals r: OP = r exactly. Points with OP less than r are inside, points with OP greater than r are outside, and neither kind belongs to the circle. Check the definition on coordinates: is (1, 7) on the circle with center (4, 3) and radius 5? The distance is the square root of (4 − 1)² + (3 − 7)² = 9 + 16 = 25, which is 5, so yes. Is (8, 6)? The square root of 16 + 9 = 25 is 5, so yes. Is (6, 6)? The square root of 4 + 9 = 13 is about 3.61, less than 5, so (6, 6) is inside the circle and not on it. The center is not a point of the circle: its distance from itself is 0, not r.

Perpendicular lines. Two lines are perpendicular when they intersect and one of the four angles they form is a right angle. Because the angles around the intersection come in supplementary pairs, one right angle forces all four to be right angles. On a coordinate grid, the line through (0, 0) and (2, 4) and the line through (2, 4) and (6, 2) are perpendicular: the first has slope 4 ÷ 2 = 2 and the second has slope (2 − 4) ÷ (6 − 2) = −2 ÷ 4 = −1/2, and slopes whose product is −1 belong to perpendicular lines. A segment can be perpendicular to a line in the same sense, by extending the segment to its line.

Parallel lines. Two distinct lines in the same plane are parallel when they have no point in common. This is the whole definition; "the same distance apart everywhere" is a consequence that gets proved later, not part of the definition. A line is not called parallel to itself under this definition, though some books allow it; say which convention you are using. On a coordinate grid, distinct non-vertical lines are parallel exactly when their slopes are equal, and two distinct vertical lines are parallel.

A worked model of using a definition. A student claims that the set of points P with AP + PB = 10, where A and B are 6 units apart, is the segment AB. Test it against the definition. The segment AB is the set of P on line AB with AP + PB = AB = 6. The student's set has AP + PB = 10, not 6, so it is a different set: it is an oval-shaped curve (an ellipse) surrounding A and B, and it contains points off the line entirely, such as the point above the midpoint of AB whose distances to A and B are both 5. Because that point is not on line AB, it cannot belong to the segment. The claim fails on two counts, membership of points off the line and the wrong constant.

The misconception. Students often think a definition is just a description, so any true sentence about circles counts as a definition of a circle. "A circle is round" is true and useless: an ellipse is also round. A definition must pin the object down completely, which is why "the set of all points at distance r from O" works and "round" does not. Precision is what lets later proofs use a definition as a two-way tool: if P is on the circle then OP = r, and if OP = r then P is on the circle.

Worked, circle membership: Decide whether (0, 0), (4, 8) and (5, 6) lie on the circle with center (4, 3) and radius 5. For (0, 0): the distance is the square root of 4² + 3² = 25, which is 5, so (0, 0) is on the circle. For (4, 8): the square root of 0² + 5² = 5, on the circle. For (5, 6): the square root of 1² + 3² = 10, about 3.16, less than 5, so (5, 6) is inside the circle and is not a point of it. The definition decides each case by one distance computation.

Worked, segment or ray: Points A(2, 1) and B(6, 1) are given, along with P(9, 1) and Q(4, 1). Both P and Q are on line AB, which is the horizontal line y = 1. For Q: AQ = 2 and QB = 2, so AQ + QB = 4 = AB, and Q is on segment AB. For P: AP = 7 and PB = 3, so AP + PB = 10, not 4; P is not on the segment. Is P on ray AB? Ray AB contains the points of the line for which A is not between the point and B. Between P and B the sum PB + BA = 3 + 4 = 7 = PA, which shows B is between A and P, not A between P and B, so P is on ray AB. It is on ray AB but not on segment AB.

Worked, angle measure from arc length: Two rays from vertex V cut an arc of length 4π units from a circle of radius 12 centered at V. The circumference of that circle is 2π × 12 = 24π. The angle measure is 360° × (4π ÷ 24π) = 360° × (1/6) = 60°. Repeating with the circle of radius 6 centered at V, the same angle cuts an arc of length 2π from a circumference of 12π, and 360° × (2π ÷ 12π) = 60° again. The measure is a property of the angle, not of the circle used to measure it.

Worked, perpendicular or not: Line p passes through (1, 2) and (4, 8); line q passes through (4, 8) and (10, 5). Slope of p: (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2. Slope of q: (5 − 8) ÷ (10 − 4) = −3 ÷ 6 = −1/2. The product is 2 × (−1/2) = −1, so the lines are perpendicular at their common point (4, 8). Had q passed through (10, 6) instead, its slope would be −2 ÷ 6 = −1/3, the product would be −2/3 and the lines would meet at an angle other than 90°.

Recording task: Make a two-column table with the six defined terms of this lesson (segment, ray, angle, circle, perpendicular lines, parallel lines) in the left column. In the right column write each definition in your own words using only the undefined notions and terms defined above it in the table, then underline every undefined notion you used. Below the table, draw on grid paper the circle with center (4, 3) and radius 5, mark these seven of the twelve grid points that lie on it, (0, 0), (1, 7), (7, 7), (8, 6), (4, 8), (4, −2) and (9, 3), and show the distance computation for two of them.

Independent performance: Write a faulty definition for each of three terms, for example "a circle is a set of points the same distance from a center" (which omits that all such points are included), and for each one produce a specific object that satisfies your faulty wording but is not the intended figure. Then repair each definition. Success criteria: each faulty definition is accompanied by a concrete counterexample, with coordinates or a labeled sketch; each repaired definition uses only undefined notions and previously defined terms; each repaired definition includes every intended object and excludes every counterexample you produced; and a closing paragraph explains, with the circle definition as the example, why a definition must work in both directions.