The angle bisector; sum and difference of angles

A ray that starts at the vertex of an angle and goes through it splits the angle into two angles, and the whole angle is their sum. In the figure, the ray $\displaystyle BD$ splits $\displaystyle \angle ABC$ into two angles: $\displaystyle \angle ABD$ and $\displaystyle \angle DBC$.

The two angles together fill $\displaystyle \angle ABC$, so:

Taking one of the parts away from the whole angle leaves the other part:

In the figure above, $\displaystyle {\angle ABD}={35^{\circ}}$ and $\displaystyle {\angle DBC}={50^{\circ}}$.

• The whole angle: $\displaystyle {\angle ABC}={35^{\circ}+50^{\circ}}={85^{\circ}}$.
• The other way round: when we know that $\displaystyle {\angle ABC}={85^{\circ}}$ and $\displaystyle {\angle ABD}={35^{\circ}}$, we subtract and get $\displaystyle {\angle DBC}={85^{\circ}-35^{\circ}}={50^{\circ}}$.

An angle bisector is a ray that starts at the vertex and splits the angle into two equal angles. In the figure, $\displaystyle BD$ bisects $\displaystyle \angle ABC$. The same mark on both arcs means that the two angles are equal.

Each of the two angles is half of the whole angle:

• For example: when $\displaystyle {\angle ABC}={70^{\circ}}$, each of the angles $\displaystyle \angle ABD$ and $\displaystyle \angle DBC$ is $\displaystyle {70^{\circ}:2}={35^{\circ}}$.
• The angle bisector can be found without a protractor, too: cut an angle out of paper and fold it so that its arms lie on each other. The fold is the angle bisector.

In the figure, $\displaystyle {\angle ABD}={28^{\circ}}$ and $\displaystyle {\angle DBC}={46^{\circ}}$.
Calculate $\displaystyle \angle ABC$.

In the figure, $\displaystyle {\angle ABC}={130^{\circ}}$ and $\displaystyle {\angle ABD}={55^{\circ}}$.
Calculate $\displaystyle \alpha$.

$\displaystyle BD$ bisects $\displaystyle \angle ABC$.
$\displaystyle {\angle ABC}={64^{\circ}}$.
Calculate $\displaystyle \angle ABD$.

$\displaystyle BD$ bisects $\displaystyle \angle ABC$.
$\displaystyle {\angle DBC}={38^{\circ}}$.
Calculate $\displaystyle \angle ABC$.

$\displaystyle \angle ABC$ is a right angle.
$\displaystyle {\angle ABE}={25^{\circ}}$ and $\displaystyle {\angle DBC}={30^{\circ}}$.
Calculate $\displaystyle \alpha$.

The points $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are on one straight line, with $\displaystyle B$ between them.
$\displaystyle BD$ bisects $\displaystyle \angle ABC$.
What is the size of $\displaystyle \angle ABD$, and what kind of angle is it? Explain.

$\displaystyle BD$ bisects $\displaystyle \angle ABC$, and $\displaystyle BE$ bisects $\displaystyle \angle DBC$.
$\displaystyle {\angle ABC}={80^{\circ}}$.
Calculate $\displaystyle \angle EBC$.