Question

what is the measure of \\( \angle dae \\)?
\\( 45 ^ { \circ } \\)
\\( 46 ^ { \circ } \\)
\\( 91 ^ { \circ } \\)
\\( 146 ^ { \circ } \\)

Explanation:

Step1: Find the sum of angles around a point

The sum of angles around a point \(A\) is \(360^{\circ}\).

Step2: Use the given angles

We know two angles: \(\angle CAD = 89^{\circ}\) and \(\angle BAF=134^{\circ}\). Since \(\angle BAF\) and \(\angle CAF\) are vertical - opposite angles, \(\angle CAF = 134^{\circ}\).

Step3: Calculate \(\angle DAE\)

Let \(\angle DAE=x\). Then \(89^{\circ}+134^{\circ}+x + 134^{\circ}+89^{\circ}=360^{\circ}\) (sum of angles around a point). But a better way:
We know that \(\angle CAD+\angle DAE+\angle EAF+\angle FAB+\angle BAC = 360^{\circ}\), but also, since \(\angle BAC\) and \(\angle FAE\) are vertical - opposite angles.
Another approach:
The sum of angles around point \(A\): \(89^{\circ}+134^{\circ}+\angle DAE + 134^{\circ}+89^{\circ}=360^{\circ}\) (not the most efficient).
The most efficient:
Since \(\angle CAD = 89^{\circ}\), \(\angle BAF = 134^{\circ}\), and \(\angle BAF\) and \(\angle CAF\) are vertical angles (\(\angle CAF = 134^{\circ}\)).
We know that \(\angle CAD+\angle DAE+\angle EAF = 180^{\circ}\) (a straight line). But \(\angle EAF=\angle BAC\) (vertical angles).
The sum of angles around \(A\):
\(\angle CAD+\angle DAE+\angle EAF+\angle FAB+\angle BAC = 360^{\circ}\), but using the fact that \(\angle BAC\) and \(\angle FAE\) are equal.
The correct way:
Since \(\angle CAD = 89^{\circ}\), \(\angle BAF = 134^{\circ}\), and \(\angle BAF\) and \(\angle CAF\) are vertical angles (\(\angle CAF=134^{\circ}\))
We use the formula \(\angle DAE=360^{\circ}- 2\times(89^{\circ}+134^{\circ})\div2\) (alternatively)
The sum of \(\angle CAD\) and \(\angle BAF\) and their vertical - opposite angles: \(2\times(89^{\circ}+134^{\circ})\). Then \(\angle DAE=\frac{360^{\circ}-2\times(89^{\circ}+134^{\circ})}{2}\)

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Or, using the property that \(\angle CAD+\angle DAE+\angle EAF = 180^{\circ}\) (assuming a straight - line - like property after considering vertical angles).
Let's assume a full - circle \(360^{\circ}\). The non - \(\angle DAE\) part: \(89^{\circ}+134^{\circ}+134^{\circ}+89^{\circ}=446^{\circ}\) (counting twice). But actually, \(\angle DAE=360^{\circ}-(89^{\circ}+134^{\circ}+134^{\circ}+89^{\circ})\div2\)

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Answer:

45°