Question

4 multiple choice 25 points find the value of x. encuentra el valor de x. options: 35°, 22°, 101°, 123°

Explanation:

Step1: Find the interior angle at D

The angle adjacent to \(123^\circ\) (angle \( \angle EDJ\)) and the interior angle at \(D\) (let's call it \( \angle EDF\)) are supplementary. So, \( \angle EDF = 180^\circ - 123^\circ = 57^\circ\)? Wait, no, wait. Wait, in triangle \(EDF\), we know angle at \(E\) is \(22^\circ\), angle at \(D\) (interior) and we need to find angle at \(F\) (which is \(x\)). Wait, no, maybe I made a mistake. Wait, the exterior angle at \(D\) is \(123^\circ\)? Wait, no, the angle marked \(123^\circ\) is adjacent to the interior angle of the triangle at \(D\). So the interior angle at \(D\) is \(180^\circ - 123^\circ = 57^\circ\)? Wait, no, that can't be. Wait, no, the exterior angle theorem: the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Wait, the angle \( \angle EDJ = 123^\circ\) is an exterior angle to triangle \(EDF\), with respect to vertex \(D\). So, by exterior angle theorem, \( \angle EDJ = \angle E + \angle F\). So \(123^\circ = 22^\circ + x\). Then, solving for \(x\): \(x = 123^\circ - 22^\circ = 101^\circ\)? Wait, no, that's not right. Wait, no, maybe the interior angle at \(D\) is \(180 - 123 = 57\), then in triangle, sum of angles is \(180\). So \(22 + 57 + x = 180\). Then \(x = 180 - 22 - 57 = 101\). Wait, but let's check again.

Wait, the diagram: point \(D\), with a line \(DJ\) going down, so angle \( \angle EDF\) and \( \angle EDJ\) are supplementary. So \( \angle EDF = 180^\circ - 123^\circ = 57^\circ\). Then in triangle \(EDF\), angles sum to \(180^\circ\). So \( \angle E + \angle D + \angle F = 180^\circ\). So \(22^\circ + 57^\circ + x = 180^\circ\). Then \(x = 180 - 22 - 57 = 101^\circ\)? Wait, but the options have \(101^\circ\) as an option. Wait, but let's check the exterior angle theorem again. The exterior angle at \(D\) (which is \(123^\circ\)) should be equal to the sum of the two remote interior angles (angles at \(E\) and \(F\)). So \(123^\circ = 22^\circ + x\). Then \(x = 123 - 22 = 101^\circ\). Yes, that's correct. So the value of \(x\) is \(101^\circ\).

Step1: Apply Exterior Angle Theorem

The exterior angle (\(123^\circ\)) of triangle \(EDF\) at \(D\) equals the sum of the two non - adjacent interior angles (\(22^\circ\) and \(x^\circ\)). So, we have the equation:
\[123 = 22 + x\]

Step2: Solve for \(x\)

Subtract \(22\) from both sides of the equation:
\[x = 123 - 22\]
\[x = 101\]

Answer:

\(101^\circ\) (corresponding to the option with \(101^\circ\))