Question

analyzing angle relationships
the measure of $\angle mkj$ is $80^\circ$. after analyzing the diagram, janelle concludes that $\overleftrightarrow{kl}$ is an angle bisector.

which statement best describes janelle’s conclusion?
\bigcirc her conclusion is incorrect because $2x + 10$ is not equal to $3x - 5$.
\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 $m\angle lkm + m\angle lkj = m\angle mkj$.

Explanation:

Step1: Recall angle bisector definition

An angle bisector divides an angle into two equal angles. So, if \(\overrightarrow{KL}\) is an angle bisector, then \(m\angle LKM = m\angle LKJ\), which means \(2x + 10=3x - 5\) and also \(m\angle LKM + m\angle LKJ=m\angle MKJ = 80^\circ\).

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

Subtract \(2x\) from both sides: \(10=x - 5\). Then add 5 to both sides: \(x = 15\).

Step3: Check if the sum is \(80^\circ\)

Substitute \(x = 15\) into \(2x + 10\) and \(3x - 5\).
For \(2x+10\): \(2(15)+10 = 30 + 10=40^\circ\).
For \(3x - 5\): \(3(15)-5=45 - 5 = 40^\circ\).
Now, sum the two angles: \(40^\circ+40^\circ = 80^\circ\), which matches \(m\angle MKJ\). So the angles are equal and their sum is \(80^\circ\), so \(\overrightarrow{KL}\) is an angle bisector.

Step4: Analyze each option

• Option 1: Incorrect, because \(2x + 10 = 3x - 5\) gives \(x = 15\) (as shown), so they are equal when \(x = 15\).
• Option 2: Incorrect, we can use the angle sum and the bisector definition to draw a conclusion.
• Option 3: Correct, because \(x = 15\) makes the two angles equal (\(40^\circ\) each) and their sum is \(80^\circ\), so \(\overrightarrow{KL}\) bisects \(\angle MKJ\).
• Option 4: The reason is not about the angle - sum formula for the conclusion, the key is that the two angles are equal (from \(x = 15\)) which makes it a bisector. But option 3's reasoning is correct as \(x = 15\) shows the angles are equal.

Answer:

Her conclusion is correct because the value of \(x\) is 15.