Given::
[tex]\begin{gathered} \angle4=52^{\circ} \\ \angle6=90^{\circ} \end{gathered}[/tex]
First, we find:
[tex]\angle3\text{ and }\angle5[/tex]
Since, the vertically opposite angles are equal.
Therefore,
[tex]\begin{gathered} \angle3=\angle6 \\ \therefore\angle3=90^{\circ} \end{gathered}[/tex]
Next to find the angle of 5:
We have,
[tex]\begin{gathered} \angle1=\angle4 \\ \therefore\angle1=52^{\circ} \\ \angle3=90^{\circ} \\ \angle4=52^{\circ} \\ \angle6=90^{\circ} \end{gathered}[/tex]
Since the angle of 2 and 5 are vertical angles and we know that the central angle is 360.
So that,
[tex]\begin{gathered} \angle1+\angle2+\angle3+\angle4+\angle5+\angle6=360 \\ 52^{\circ}+\angle2+90^{\circ}+52^{\circ}+\angle5+90^{\circ}=360^{\circ} \\ \angle2+\angle5=360-284 \\ \angle5+\angle5=76 \\ 2\angle5=76 \\ \angle5=38^{\circ} \end{gathered}[/tex]
Hence, the measures of angle of 3 and 5 are,
[tex]\angle3=90^{\circ}\text{ and }\angle5=38^{\circ}[/tex]
Finally, to find the complementary pair of angle 1.
[tex]\begin{gathered} \angle1=52^{\circ} \\ 52^{\circ}+38^{\circ}=90^{\circ} \\ \therefore\angle1+\angle2=90^{\circ} \\ \therefore\angle1+\angle5=90^{\circ} \end{gathered}[/tex]
Hence, the two angles which are complementary to the angle of 1 is,
[tex]\angle2\text{ and }\angle5[/tex]
