Question

which statement regarding the diagram is true?
○ ( mangle mkl + mangle mlk = mangle jkm )
○ ( mangle kml + mangle mlk = mangle jkm )
○ ( mangle mkl + mangle mlk = 180^circ )
○ ( mangle jkm + mangle mlk = 180^circ )

Explanation:

Step1: Recall the Exterior Angle Theorem

The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In triangle \(MKL\), \(\angle JKM\) is an exterior angle at vertex \(K\), and the two non - adjacent interior angles are \(\angle KML\) and \(\angle MLK\)? Wait, no, wait. Wait, the interior angles at \(K\) and \(L\) and \(M\). Wait, \(\angle MKL\) and \(\angle MLK\) are two interior angles, and \(\angle JKM\) is adjacent to \(\angle MKL\) (they are supplementary? No, wait, \(J - K - L\)? No, \(J\), \(K\) are on a straight line, so \(\angle JKM\) and \(\angle MKL\) are supplementary (\(m\angle JKM + m\angle MKL=180^{\circ}\))? Wait, no, let's re - examine.

Wait, the triangle is \( \triangle MKL\). The side \(JK\) is a straight line extension of \(MK\)? Wait, \(J\), \(K\) are colinear, so \(\angle JKM\) is an exterior angle of \(\triangle MKL\) at \(K\). The two non - adjacent interior angles to \(\angle JKM\) are \(\angle KML\) and \(\angle MLK\)? No, wait, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non - adjacent) interior angles.

Wait, in \(\triangle MKL\), angle at \(K\): \(\angle MKL\), angle at \(L\): \(\angle MLK\), angle at \(M\): \(\angle KML\). The exterior angle at \(K\) ( \(\angle JKM\)) should be equal to the sum of the two non - adjacent interior angles, which are \(\angle KML\) and \(\angle MLK\)? Wait, no, wait. Wait, \(\angle MKL\) and \(\angle JKM\) are supplementary (since \(J\), \(K\), \(L\)? No, \(J\), \(K\) are on a line, so \(\angle JKM + \angle MKL = 180^{\circ}\). But the exterior angle theorem: for \(\triangle MKL\), the exterior angle at \(K\) ( \(\angle JKM\)) is equal to \(\angle KML+\angle MLK\). Let's check the options:

Option 1: \(m\angle MKL + m\angle MLK=m\angle JKM\) → No, because \(\angle MKL\) is adjacent to \(\angle JKM\), not non - adjacent.

Option 2: \(m\angle KML + m\angle MLK=m\angle JKM\) → By the Exterior Angle Theorem, this is correct. Let's check other options.

Option 3: \(m\angle MKL + m\angle MLK = 180^{\circ}\) → In a triangle, the sum of two interior angles is not necessarily \(180^{\circ}\) (the sum of all three is \(180^{\circ}\)), so this is wrong.

Option 4: \(m\angle JKM + m\angle MLK=180^{\circ}\) → There is no reason for these two angles to be supplementary, so this is wrong.

Answer:

\(m\angle KML + m\angle MLK = m\angle JKM\) (the second option)