Question

analyzing angle relationships
the measure of \\( \angle mkj \\) is \\( 80 ^ { \circ } \\). after analyzing the

• diagram, janelle concludes that \\( \overline { k l } \\) is an angle bisector.

which statement best describes janelles conclusion?
\\( \bigcirc \\) her conclusion is incorrect because \\( 2 x + 10 \\) is not
equal to \\( 3 x - 5 \\).
\\( \bigcirc \\) her conclusion is incorrect because the angles are
not marked; therefore, no conclusion can be drawn.
\\( \bigcirc \\) her conclusion is correct because the value of \\( x \\) is
15.
\\( \bigcirc \\) her conclusion is correct because
\\( m \angle l k m + m \angle l k j = m \angle m k j \\).

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles. If \(KL\) were an angle bisector, then \(3x - 5=2x + 10\).

Step2: Solve the equation \(3x - 5=2x + 10\) for \(x\)

Subtract \(2x\) from both sides: \(3x-2x-5=2x - 2x+10\), which gives \(x-5 = 10\).
Add 5 to both sides: \(x=10 + 5=15\).

Step3: Find the measures of \(\angle LKM\) and \(\angle LKJ\)

If \(x = 15\), then \(m\angle LKM=(2x + 10)^{\circ}=(2\times15+10)^{\circ}=(30 + 10)^{\circ}=40^{\circ}\) and \(m\angle LKJ=(3x - 5)^{\circ}=(3\times15-5)^{\circ}=(45-5)^{\circ}=40^{\circ}\). But the measure of \(\angle MKJ=m\angle LKM + m\angle LKJ=80^{\circ}\). However, the problem does not state that \(ML\parallel KJ\) (there are no parallel - line markings). Without the information that \(ML\parallel KJ\) (so that we can use the properties of alternate - interior angles or corresponding angles to establish a relationship between the angles formed by the transversal \(KL\)), we cannot assume \(3x - 5=2x + 10\) based on the given diagram (just the angle \(\angle MKJ = 80^{\circ}\) is given).

Answer:

Her conclusion is incorrect because the angles are not marked; therefore, no conclusion can be drawn.