Points, Lines, and Angles

Points, lines, segments, and rays

Four objects do almost all the work in basic geometry. They differ by one simple question: how far do they extend?

• A point marks a single location and has no size at all. We draw it as a dot and name it with a capital letter, like point $A$.
• A line is perfectly straight and runs forever in both directions. The line through two points $A$ and $B$ is written $\overleftrightarrow{AB}$. A line has no endpoints.
• A line segment is the straight piece between two points, including both ends. The segment from $A$ to $B$ is written $\overline{AB}$. A segment has exactly two endpoints.
• A ray starts at one point and runs forever in one direction, like a beam of light from a flashlight. The ray starting at $A$ and passing through $B$ is written $\overrightarrow{AB}$. A ray has exactly one endpoint, and we always name that endpoint first.

The arrowheads in a drawing carry the meaning: an arrow says “keeps going this way,” while a plain endpoint says “stops here.”

So the count of endpoints tells the three straight objects apart: a line has $0$, a ray has $1$, and a segment has $2$. The length of a segment is a real number we can measure (the distance between its two ends). A line and a ray, by contrast, run forever, so neither has a length.

Planes, collinear points, intersecting lines

Points and lines live on a flat surface called a plane. You can picture a plane as an endless sheet of paper that extends forever in every direction. A few more words name common situations on a plane:

• Points that all lie on one straight line are collinear. Any two points are always collinear, since a single line passes through them. Three points, however, may or may not be collinear: pick three dots at random and they usually are not.
• Two different lines that cross share exactly one point; they are intersecting lines. The shared point is the only place they meet, because two straight lines cannot cross twice.

We will meet two more relationships between lines, parallel and perpendicular, after we have angles to describe them with.

Angles: two rays from a common vertex

When two rays start from the same endpoint, the opening between them is an angle. The shared endpoint is the vertex, and the two rays are the sides (or arms) of the angle.

We name an angle with the $\angle$ symbol. Using three letters, the middle letter is always the vertex. The angle below can be written $\angle ABC$ or $\angle CBA$, and either way the vertex is $B$. When only one angle sits at a vertex, we may name it by that single vertex letter, like $\angle B$.

The size of an angle measures how far one side is turned from the other. It does not depend on how long you draw the sides. Stretching the arms longer does not change the angle, because the amount of turn between them is the same. Angle size is measured in degrees, written with a small raised circle, like $90^\circ$.

Measuring angles in degrees

Imagine standing at the vertex and rotating one ray until it lands on the other. A full turn, all the way around back to the start, is defined as $360^\circ$. Everything else is a fraction of that full turn:

A half turn ($180^\circ$) leaves the ray pointing in exactly the opposite direction, forming a straight line. A quarter turn ($90^\circ$) makes a square corner, called a right angle. These two landmarks, $90^\circ$ and $180^\circ$, are worth memorizing, because every angle type is described relative to them.

The four basic types of angles

We sort angles by how their measure compares to the $90^\circ$ and $180^\circ$ landmarks. Four basic types cover every opening up to a straight line:

• An acute angle is greater than $0^\circ$ and less than $90^\circ$ (a narrow opening).
• A right angle is exactly $90^\circ$ (a square corner). We mark it with a small square instead of an arc.
• An obtuse angle is between $90^\circ$ and $180^\circ$ (a wide opening).
• A straight angle is exactly $180^\circ$: the two sides point in opposite directions and form a straight line.

A fifth type covers everything past a straight line. An angle larger than $180^\circ$ but less than $360^\circ$ is called a reflex angle. A reflex angle is the “outside” opening you get when you turn more than halfway around but not the whole way. For any opening below $180^\circ$, the reflex angle on the other side is whatever is left of the full turn. The opening and its reflex angle therefore add to $360^\circ$. For example, the reflex angle that goes with a $110^\circ$ opening is $360^\circ - 110^\circ = 250^\circ$.

Complementary and supplementary angles

Two angles are often described by what they add up to.

Two angles are complementary when their measures add to $90^\circ$. If one of them is $x$, the other (its complement) is

A complement only exists when $x$ is less than $90^\circ$, since otherwise $90^\circ - x$ would be zero or negative. For example, the complement of $30^\circ$ is $60^\circ$, because $30^\circ + 60^\circ = 90^\circ$.

Two angles are supplementary when their measures add to $180^\circ$. If one is $x$, the other (its supplement) is

A supplement only exists when $x$ is less than $180^\circ$, since otherwise $180^\circ - x$ would be zero or negative. For example, the supplement of $110^\circ$ is $70^\circ$, because $110^\circ + 70^\circ = 180^\circ$.

The two angles do not have to be drawn touching for the words to apply. Any two angles whose measures happen to add to $90^\circ$ are complementary, and any two that add to $180^\circ$ are supplementary, wherever they sit.

Adjacent angles and linear pairs

Two angles are adjacent when they share a vertex and a side but do not overlap. Adjacent angles sit right next to each other, like two slices of the same pie.

A special case is the linear pair: two adjacent angles whose outer sides point in opposite directions, together forming a straight line. Because a straight line is a straight angle of $180^\circ$, the two angles of a linear pair must add to $180^\circ$. In other words, a linear pair is always supplementary. This single fact is the key to the next result.

Vertical angles are equal

When two straight lines cross, they make four angles at the crossing point. The angles directly across from each other (sharing only the vertex, not a side) are called vertical angles, and they come in two pairs. The headline fact is that vertical angles are always equal.

In the picture, the two angles labeled $a$ are a vertical pair and are equal. The two angles labeled $b$ are the other vertical pair, and they are equal as well.

Why must this be true? It follows directly from linear pairs.

The proof works for any two lines that cross, so test it on as many crossings as you like. Drag point $A$ or point $B$ to turn either line, and watch angles $1$ and $3$: however you turn the lines, they stay equal, and so do angles $2$ and $4$.

Angles around a point

If several angles share a vertex and together sweep all the way around it with no gaps or overlaps, they make one full turn. So angles around a point add to $360^\circ$. For instance, three angles meeting at a point with no gaps, measuring $120^\circ$, $150^\circ$, and an unknown $x$, must satisfy

which gives $x = 90^\circ$. This is just the full-turn idea from before: one complete rotation is $360^\circ$, whether you split it into two pieces, four pieces, or any number.

Parallel and perpendicular lines

Now that we can measure angles, two more relationships between lines have clean definitions.

Two lines in the same plane that never meet, staying the same distance apart forever, are parallel. Think of the two rails of a straight train track. We write $\ell \parallel m$ to say line $\ell$ is parallel to line $m$.

Two lines that cross to form a right angle are perpendicular, written $\ell \perp m$. Because vertical angles are equal and a linear pair is supplementary, one $90^\circ$ angle at the crossing makes the other three $90^\circ$ too. The angle across from that right angle is its equal vertical angle ($90^\circ$), and each neighbor is its supplement ($180^\circ - 90^\circ = 90^\circ$). So perpendicular lines meet at four right angles, and we only need to mark one of them with the small square. The corner of this page and the cross of a plus sign are both perpendicular.

Parallel lines and a transversal

A line that crosses two (or more) other lines is called a transversal. When the two lines it crosses are parallel, the angles formed line up in tidy, predictable ways. Three facts come up again and again, and each one has a clear reason behind it.

Corresponding angles (the ones in matching positions, such as the upper-right angle at each crossing, both labeled $x$ below) are equal. Here is why. The two lines are parallel, which means they run in exactly the same direction. The transversal is a single straight line, so it keeps the same direction the whole way down. When the same slanted line meets two lines that point the same way, it has to open up the same amount at each meeting. The angle cannot change if nothing about the directions changed. A clean way to see it is to slide the lower crossing straight up along the transversal. Parallel lines stay the same distance apart, so the slide lands the lower parallel line exactly on the upper line. The transversal slides along itself, so that single slide carries the whole lower picture onto the whole upper picture. The angle in each position lands on the matching angle, so they must be equal.

Alternate interior angles (on opposite sides of the transversal, in the space between the two parallel lines) are equal. We do not need a new idea for this one: it follows from corresponding angles plus vertical angles. Take an interior angle on one side of the transversal. The corresponding angle at the other crossing is equal to it (corresponding angles). That corresponding angle and the alternate interior angle we are after are vertical angles at that crossing. They are the opposite pair where the transversal meets the line, so they are equal too. Equal to the same angle means equal to each other, so the two alternate interior angles match.

Co-interior angles (also called same-side interior angles: both between the parallel lines, both on the same side of the transversal) are supplementary, adding to $180^\circ$. This one we can derive as well. Look at one crossing. A co-interior angle has an interior angle right beside it, on the same line, on the other side of the transversal. Those two angles form a linear pair, so they add to $180^\circ$. But that neighbor is the alternate interior angle of the co-interior angle at the other crossing, and we just showed alternate interior angles are equal. Replacing the neighbor with its equal partner shows the two co-interior angles add to $180^\circ$.

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