## Task 3 - 330833

For every triangle $ABC$, the following notations are introduced:

The line through $A$ and $B$ is $u$, the line through $B$ and $C$ is $v$, the line through $C$ and $A$ is $w$. When $u$ is reflected across $v$, it becomes the line $p$. When $u$ is reflected across $w$, it becomes the line $q$.

As usual, the measures of the interior angles $\angle BAC$ and $\angle ABC$ are denoted by $\alpha$ and $\beta$, respectively. For the following tasks, it is always assumed that $\alpha = 55^\circ$.

a) At what angle do the lines $p$ and $q$ intersect if $\beta = 75^\circ$?

b) How large must $\beta$ be for $p$ and $q$ to be perpendicular to each other?

c) Give a value of $\beta$ such that $p$ and $q$ can be shown to be parallel to each other!

d), e) Create a drawing in which $p$ and $q$ intersect at a point on the same side of the line $u$ as $C$; choose the triangle $ABC$

d) with an acute interior angle at $B$.

e) with an obtuse interior angle at $B$.

(For d) and e), no justification is required.)
a) One obtains $p$ by laying off the angle of size $\beta=75^{\circ}$ on the side of segment $BC$ at point $B$ where $A$ does not lie.

Similarly, one obtains $q$ by laying off the angle of $55^{\circ}$ on $AC$ (see figure).

For the intersection point $D$ of $p$ and $q$, it follows that $\angle ABD = 180^{\circ} - 2 \cdot 75^{\circ} = 30^{\circ}$ and $\angle BAD = 180^{\circ} - 2 \cdot 55^{\circ} = 70^{\circ}$.

By the angle sum theorem for triangle $ABD$, it follows that $p$ and $q$ intersect at an angle of size $\delta = \angle ADB = 180^{\circ} - 30^{\circ} - 70^{\circ} = 80^{\circ}$.

b) As in a), $\angle ABD = 180^{\circ} - 2\beta$, $\angle BAD = 70^{\circ}$, so $\delta = 180^{\circ} - (180^{\circ} - 2\beta) - 70^{\circ}$ (see 2nd figure).

Since $\delta = 90^{\circ}$ is required, it follows that $2\beta = 90^{\circ} + 70^{\circ}$, thus $\beta = 80^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0830.jpg?height=434&width=491&top_left_y=2190&top_left_x=1294)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0831.jpg?height=277&width=491&top_left_y=244&top_left_x=246)

c) For $\beta = 35^{\circ}$, $p$ and $q$ each form an angle of $2 \cdot 35^{\circ}$ and $2 \cdot 55^{\circ}$, respectively, on the side where $C$ lies (see 3rd figure). Since these angles sum to $180^{\circ}$, $p$ and $q$ are parallel for the given value $\beta = 35^{\circ}$.

d), e) The figures show a drawing of the required type.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0831.jpg?height=412&width=440&top_left_y=545&top_left_x=1136)