Question

Find the measures of <5, <3, and which two angles are complementary to <1?

Answer 1

Given::

`\begin{gathered} \angle4=52^(\circ) \\ \angle6=90^(\circ) \end{gathered}`

First, we find:

`\angle3\text{ and }\angle5`

Since, the vertically opposite angles are equal.

Therefore,

`\begin{gathered} \angle3=\angle6 \\ \therefore\angle3=90^(\circ) \end{gathered}`

Next to find the angle of 5:

We have,

`\begin{gathered} \angle1=\angle4 \\ \therefore\angle1=52^(\circ) \\ \angle3=90^(\circ) \\ \angle4=52^(\circ) \\ \angle6=90^(\circ) \end{gathered}`

Since the angle of 2 and 5 are vertical angles and we know that the central angle is 360.

So that,

`\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}`

Hence, the measures of angle of 3 and 5 are,

`\angle3=90^(\circ)\text{ and }\angle5=38^(\circ)`

Finally, to find the complementary pair of angle 1.

`\begin{gathered} \angle1=52^(\circ) \\ 52^(\circ)+38^(\circ)=90^(\circ) \\ \therefore\angle1+\angle2=90^(\circ) \\ \therefore\angle1+\angle5=90^(\circ) \end{gathered}`

Hence, the two angles which are complementary to the angle of 1 is,

`\angle2\text{ and }\angle5`