Question
In the following diagram
\[\overline{DE} \parallel \overline{FG}\] and
\[\overline{KL} \perp \overline{FG}\].
What is the measure of
\[\blueD{\angle x}\]?
Angles are not necessarily drawn to scale. Given the following:
Two parallel lines labeled line DE and line FG. Line KL intersects the parallel lines at points A and B. Line IJ intersects the parallel lines at points A and C. Angle IAK is 62 degrees. Angle GCJ is x degrees.
\[D\]
\[E\]
\[F\]
\[G\]
\[I\]
\[J\]
\[K\]
\[L\]
\[A\]
\[B\]
\[C\]
\[\greenD{62^\circ}\]
\[\blueD{x^\circ}\]
\[\blueD{\angle{x}} = \]
\[\Large{{}^\circ}\]
\[\overline{DE} \parallel \overline{FG}\] and
\[\overline{KL} \perp \overline{FG}\].
What is the measure of
\[\blueD{\angle x}\]?
Angles are not necessarily drawn to scale. Given the following:
Two parallel lines labeled line DE and line FG. Line KL intersects the parallel lines at points A and B. Line IJ intersects the parallel lines at points A and C. Angle IAK is 62 degrees. Angle GCJ is x degrees.
\[D\]
\[E\]
\[F\]
\[G\]
\[I\]
\[J\]
\[K\]
\[L\]
\[A\]
\[B\]
\[C\]
\[\greenD{62^\circ}\]
\[\blueD{x^\circ}\]
\[\blueD{\angle{x}} = \]
\[\Large{{}^\circ}\]
Answers
GPT-4o mini
To determine the measure of \(\angle x\), we can use the properties of parallel lines and the angles formed by the transversal line \(KL\).
We're given the following:
1. \(\overline{DE} \parallel \overline{FG}\)
2. \(\overline{KL} \perp \overline{FG}\)
3. \(\angle IAK = 62^\circ\)
4. \(\angle GCJ = x^\circ\)
First, identify the relationship between the angles:
Since \(\overline{KL} \perp \overline{FG}\), angles formed where \(KL\) intersects \(FG\) are right angles. This means that \(\angle AKB\) and \(\angle BKC\) formed at point \(A\) must be right angles (i.e., \(90^\circ\)).
Also, because \(\overline{DE} \parallel \overline{FG}\) and \(IJ\) intersects them, we can apply the concept of corresponding angles.
The angle \(\angle IAK = 62^\circ\) corresponds to \(\angle GCJ\) (since they lie on the same side of transversal \(IJ\) and both are formed by intersecting transversals with the parallel lines).
Therefore, the measure of \(\angle GCJ\) can be found as follows:
\[
\angle GCJ = \angle IAK = 62^\circ
\]
This leads us to conclude:
\[
x = 62^\circ
\]
Thus, the measure of \(\blueD{\angle x}\) is
\[
\boxed{62}
\] degrees.
We're given the following:
1. \(\overline{DE} \parallel \overline{FG}\)
2. \(\overline{KL} \perp \overline{FG}\)
3. \(\angle IAK = 62^\circ\)
4. \(\angle GCJ = x^\circ\)
First, identify the relationship between the angles:
Since \(\overline{KL} \perp \overline{FG}\), angles formed where \(KL\) intersects \(FG\) are right angles. This means that \(\angle AKB\) and \(\angle BKC\) formed at point \(A\) must be right angles (i.e., \(90^\circ\)).
Also, because \(\overline{DE} \parallel \overline{FG}\) and \(IJ\) intersects them, we can apply the concept of corresponding angles.
The angle \(\angle IAK = 62^\circ\) corresponds to \(\angle GCJ\) (since they lie on the same side of transversal \(IJ\) and both are formed by intersecting transversals with the parallel lines).
Therefore, the measure of \(\angle GCJ\) can be found as follows:
\[
\angle GCJ = \angle IAK = 62^\circ
\]
This leads us to conclude:
\[
x = 62^\circ
\]
Thus, the measure of \(\blueD{\angle x}\) is
\[
\boxed{62}
\] degrees.