## Subject 3.

In the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}, O$ is the center of the face $C D D^{\prime} C^{\prime}$ and $M$ is the midpoint of the edge $B^{\prime} C^{\prime}$.

a. Find the measure of the angle between the lines $B O$ and $A D^{\prime}$.

b. Show that (AO is the bisector of the angle $\Varangle M A D$).
## Subject 3.

a. $B C^{\prime} \| A D^{\prime} \Rightarrow m \angle\left(B O, A D^{\prime}\right)=m \angle\left(B O, B C^{\prime}\right)=m\left(\angle O B C^{\prime}\right)$

..1p.

$B O$ is the median in the equilateral triangle $B D C^{\prime}$, so $m\left(\angle O B C^{\prime}\right)=30^{\circ}$

...1p.

b. Working on the right trapezoid DAMC'

Proving that $\triangle M O A$ is a right triangle (denote the side of the cube by $a$ and through calculations and the converse of the Pythagorean Theorem)

...2p.

$\operatorname{tg}(\angle M A O)=\frac{\sqrt{2}}{2}$ and $\operatorname{tg}(\angle D A O)=\frac{\sqrt{2}}{2}$

the angles are congruent, so ( $A O$ is the bisector of the angle $\angle M A D$ )

...1p.