Angles

An angle is two rays that start at the same point:

• A ray is a straight line that starts at a point and goes on without end in one direction.
• The point the rays start from is the vertex of the angle: here the point $\displaystyle B$.
• The two rays are the arms of the angle: here $\displaystyle BA$ and $\displaystyle BC$.

Angles are everywhere: between the two hands of a clock, between the blades of open scissors, between an open door and the wall.

There are three ways to name an angle:

• Three letters, the vertex in the middle: $\displaystyle \angle ABC$ (read: angle $\displaystyle ABC$). The first and the last letter are points on the arms. $\displaystyle \angle CBA$ is the same angle.
• One letter, the vertex: $\displaystyle \angle B$. Only when there is just one angle at that vertex.
• A Greek letter inside the angle: $\displaystyle \alpha$ (alpha), $\displaystyle \beta$ (beta), $\displaystyle \gamma$ (gamma), $\displaystyle \delta$ (delta).

In the figure, three rays start at the point $\displaystyle B$:

• $\displaystyle \alpha$ is $\displaystyle \angle ABD$, and $\displaystyle \beta$ is $\displaystyle \angle DBC$.
• $\displaystyle \angle ABC$ is the angle between $\displaystyle BA$ and $\displaystyle BC$. The ray $\displaystyle BD$ goes through it.
• There are three angles at the vertex $\displaystyle B$, so the name $\displaystyle \angle B$ does not say which of them is meant.

The kind of an angle depends on how wide it opens. We compare it with a right angle – like the corner of a sheet of paper or of a table:

• $\displaystyle \alpha$ – an acute angle: smaller than a right angle.
• $\displaystyle \beta$ – a right angle. It is marked with a small square instead of an arc.
• $\displaystyle \gamma$ – an obtuse angle: bigger than a right angle, but its arms do not yet make a straight line.
• $\displaystyle \delta$ – a straight angle: its two arms make one straight line.

An angle is bigger when its arms open wider. How long the arms are in the drawing makes no difference: each arm is a ray, and it goes on without end.

• $\displaystyle \beta$ is bigger than $\displaystyle \alpha$, although its arms are shorter in the drawing.
• Scissors work the same way: small scissors and big scissors can be open at the same angle.

The figure shows the triangle $\displaystyle ABC$.
The point $\displaystyle D$ is on the side $\displaystyle BC$.
Write each marked angle with three letters.

$\displaystyle \alpha$

$\displaystyle \beta$

$\displaystyle \gamma$

Which of the marked angles can be named with one letter? Explain.

Mark every right angle of the quadrilateral $\displaystyle ABCD$ on the figure.
Write the kind of each of its four angles.

What kind of angle do the two hands of a clock make at each time?

At three o’clock (3:00)

At one o’clock (1:00)

At five o’clock (5:00)

At six o’clock (6:00)

Which angle is bigger: $\displaystyle \angle ABC$ or $\displaystyle \angle DEF$?
Explain.