Mathematics · Plane geometry · Grade 7

Point, segment, angle

When two streets cross, why are opposite angles always equal?

The situation

On a city map two streets cross — four angles are formed, and for some reason opposite angles are always equal. The distance between home and school is the length of a segment, and the shop is on the road between them. A railway crosses the street at a right angle. Geometry begins with exactly these simple things: a point, a line, a segment, an angle. How do we measure and compare them, and how do we prove that “opposite angles are equal”?

Topic

The beginnings of geometry: point, line, segment, angle, adjacent and vertically opposite angles, the perpendicular. Streets on a map and a railway crossing: measuring segments, comparing angles, adjacent and vertically opposite angles

Key ideas

Segment and angle
Adjacent and vertically opposite angles
Perpendicular

Formulas

A C + C B = A B

The length of a segment is the sum of the lengths of its parts (when point $C$ lies on the segment $A B$ ).

∠ A O M = ∠ M O B = \frac{1}{2} ∠ A O B

The bisector $O M$ divides the angle into two equal parts.

∠ 1 + ∠ 2 = 180^{\circ}

Adjacent angles add up to $180^{\circ}$ : they share one side, and the other two lie on one straight line.

∠ 1 = ∠ 3

Vertically opposite angles are equal: both of them add up with $∠ 2$ to $180^{\circ}$ .

a ⟂ b \Leftrightarrow ∠ (a, b) = 90^{\circ}

Perpendicular lines cross at a right angle; the distance from a point to a line is the length of the perpendicular.

Theory

Point, line, plane

The simplest figures of geometry — the point and the straight line — are not defined; their properties are given by axioms: through any two points there passes one and only one straight line. It follows that two different straight lines have at most one common point (if they had two common points, they would be one line). All figures lie in a plane (plane geometry). The part of a straight line going from a point in one direction is a ray, the part between two points is a segment. On the first step, drag the points: the straight line always passes through them, and a second line meets it in at most one point.

Measuring and comparing segments

The length of a segment is measured in a unit of length (cm, m, km): the zero of the ruler is placed at $A$ , and the number opposite $B$ is the length $A B$ . If point $C$ lies on the segment $A B$ , then $A C + C B = A B$ . Two segments are compared by laying one on the other: the one that stays inside is the smaller. The point that divides a segment into two equal parts is the midpoint: $A M = M B = \frac{1}{2} A B$ . On the second step, drag $C$ — $A C$ and $C B$ change, but their sum is always $A B$ .

Angle and degree

An angle is two rays (the sides) coming from one point (the vertex). An angle is measured in degrees: $1^{\circ}$ is $\frac{1}{180}$ of a straight angle. Protractor: its centre is at the vertex of the angle, its zero line along one side, and the other side shows the degrees on the scale. Angles are compared by laying one on the other: vertex on vertex, side along side — the angle whose second side stays inside is the smaller. A bisector is a ray that divides an angle into two equal parts.

Types of angle, adjacent and vertically opposite angles

An angle less than $90^{\circ}$ is acute, $90^{\circ}$ is a right angle, more than $90^{\circ}$ and less than $180^{\circ}$ is obtuse, and $180^{\circ}$ is a straight angle. If two angles share one side and the other two sides form a straight line, they are adjacent: their sum is $180^{\circ}$ . If the sides of one angle are extensions of the sides of the other, they are vertically opposite. Theorem: vertically opposite angles are equal. Proof: $∠ 1 + ∠ 2 = 180^{\circ}$ and $∠ 3 + ∠ 2 = 180^{\circ}$ (both pairs are adjacent), so $∠ 1 = 180^{\circ} - ∠ 2 = ∠ 3$ .

Perpendicular lines and distance

Lines that cross at a right angle are perpendicular: $a ⟂ b$ . Two lines perpendicular to the same line do not meet (both make $90^{\circ}$ with it — they are parallel). The perpendicular from a point to a line is the shortest of all the segments joining the point to the line — that is why its length is called the distance from the point to the line. On the fifth step drag $Q$ : when $Q \neq H$ , $P Q$ is always longer than $P H$ .

The logical structure of geometry, and a note on the model

Geometry is built like a chain: definition (a new concept is named) → axiom (a basic truth accepted without proof) → theorem (a statement that must be proved) → proof (a chain of reasoning based on the axioms and earlier theorems). This structure returns in Grade 10 in the systematic review of plane geometry. Limits of the model: in the drawing, angles are whole degrees (the angle of the streets changes in $5^{\circ}$ steps), and lengths are rounded to 0.5 cm; a straight line and a ray are infinite, but on the screen they are drawn to the edge; the “streets on the map” are taken as ideal straight lines.

Did you know?

Euclid's “Elements” (about 300 BC) begins with 23 definitions and 5 postulates; the first definition is: “A point is that which has no part.”
The division of the full circle into 360 parts is thought to be inherited from the base-60 number system of the Babylonian astronomers. Another unit of angle is the radian: the full circle is $2 π \approx 6.28$ radians.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.