The bisector of an angle is a ray originating from the vertex of the angle that divides it into two other congruent angles. For example, in the drawing below, the ray $O C$ is the bisector of the angle $\angle A O B$.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-04.jpg?height=471&width=517&top_left_y=638&top_left_x=655)

a) The difference between two consecutive but non-adjacent angles is $100^{\circ}$. Determine the angle formed by their bisectors.

Note: Remember that two angles are consecutive if they share the same vertex and at least one side, and that two angles are adjacent if they do not share any interior points.

b) In the drawing below, $D A$ is the bisector of the angle $\angle C A B$. Determine the value of the angle $\angle D A E$ given that $\angle C A B + \angle E A B = 120^{\circ}$ and $\angle C A B - \angle E A B = 80^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-04.jpg?height=434&width=699&top_left_y=1645&top_left_x=587)
Solution

a) Let $\angle B A D=2 x$ and $\angle B A C=2 y$ be the adjacent angles.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-05.jpg?height=371&width=528&top_left_y=454&top_left_x=864)

The angle between the angle bisectors is

$$
\begin{aligned}
\angle E A F & =\angle E A B-\angle F A B \\
& =x-y \\
& =\frac{2 x}{2}-\frac{2 y}{2} \\
& =\frac{\angle C A B}{2}-\frac{\angle D A B}{2} \\
& =\frac{100^{\circ}}{2} \\
& =50^{\circ}
\end{aligned}
$$

b) Let $x=\angle C A D=\angle D A B$ and $y=\angle E A B$. Then $2 x+y=120^{\circ}$ and $2 x-y=80^{\circ}$. Adding the two equations, we get $4 x=200^{\circ}$, which means $x=50^{\circ}$. Substituting this value into $2 x+y=120^{\circ}$, we have $y=120^{\circ}-2 x=120^{\circ}-100^{\circ}=20^{\circ}$. Therefore,

$$
\angle D A E=x-y=50^{\circ}-20^{\circ}=30^{\circ}
$$