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, and the two non - adjacent interior angles are \(\angle KML\) and \(\angle MLK\)? Wait, no, wait. Wait, the interior angles at \(K\) and \(L\) for the exterior angle \(\angle JKM\): Wait, \(\angle JKM\) is an exterior angle at \(K\) for triangle \(MKL\). The two non - adjacent interior angles are \(\angle KML\) and \(\angle MLK\)? Wait, no, let's label the triangle. Triangle \(MKL\) has vertices \(M\), \(K\), \(L\). The angle at \(K\) inside the triangle is \(\angle MKL\), at \(L\) is \(\angle MLK\), and at \(M\) is \(\angle KML\). The exterior angle at \(K\) (extending \(JK\) from \(K\)) is \(\angle JKM\). By the Exterior Angle Theorem, \(m\angle JKM=m\angle KML + m\angle MLK\)? Wait, no, wait the Exterior Angle Theorem: the exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. So for exterior angle \(\angle JKM\) of triangle \(MKL\), the two non - adjacent interior angles are \(\angle KML\) and \(\angle MLK\)? Wait, no, \(\angle MKL\) is adjacent to \(\angle JKM\) (they are supplementary? Wait, \(\angle JKM+\angle MKL = 180^{\circ}\) because they are linear pairs. Wait, maybe I made a mistake. Let's re - examine.

Wait, the first option: \(m\angle MKL + m\angle MLK=m\angle JKM\). Let's think about the triangle angle sum. In triangle \(MKL\), \(m\angle KML + m\angle MKL + m\angle MLK=180^{\circ}\). And \(\angle JKM\) and \(\angle MKL\) are linear pairs, so \(m\angle JKM + m\angle MKL=180^{\circ}\), which means \(m\angle JKM = 180^{\circ}-m\angle MKL\). From the triangle angle sum, \(m\angle KML + m\angle MLK=180^{\circ}-m\angle MKL\). But the first option says \(m\angle MKL + m\angle MLK=m\angle JKM\). Wait, no, that can't be. Wait, maybe I mixed up the angles. Wait, let's look at the options again.

Wait, the first option: \(m\angle MKL + m\angle MLK=m\angle JKM\). Let's check the other options.

Second option: \(m\angle KML + m\angle MLK=m\angle JKM\). By the Exterior Angle Theorem, the exterior angle \(\angle JKM\) is equal to the sum of the two non - adjacent interior angles of triangle \(MKL\). The two non - adjacent interior angles to \(\angle JKM\) are \(\angle KML\) (at \(M\)) and \(\angle MLK\) (at \(L\)). So \(m\angle JKM=m\angle KML + m\angle MLK\). Wait, but let's check the first option again. Wait, maybe I mislabeled the angles. Wait, \(\angle MKL\) is the angle at \(K\) inside the triangle, \(\angle MLK\) at \(L\) inside the triangle. Then, if we consider the exterior angle \(\angle JKM\), the two non - adjacent interior angles are \(\angle KML\) and \(\angle MLK\)? No, wait, \(\angle MKL\) is adjacent to \(\angle JKM\) (linear pair), so \(\angle JKM = 180^{\circ}-\angle MKL\). And from triangle angle sum, \(\angle KML+\angle MKL+\angle MLK = 180^{\circ}\), so \(\angle KML+\angle MLK=180^{\circ}-\angle MKL=\angle JKM\). Wait, so \(m\angle KML + m\angle MLK=m\angle JKM\), which is the second option. Wait, but let's check the first option: \(m\angle MKL + m\angle MLK=m\angle JKM\). Let's take an example. Suppose in triangle \(MKL\), \(\angle MKL = 50^{\circ}\), \(\angle MLK = 60^{\circ}\), then \(\angle KML=70^{\circ}\). Then \(\angle JKM = 180 - 50=130^{\circ}\). Now, \(m\angle MKL + m\angle MLK=50 + 60 = 110 eq130\). \(m\angle KML + m\angle MLK=70 + 60 = 130=m\angle JKM\). So the second option is correct? Wait, but the first option was \(m\angle MKL + m\angle MLK=m\angle JKM\), no. Wait, maybe I made a mistake in the angle labels. Wait, the first option: \(m\angle MKL + m\angle MLK=m\angle JKM\). Wait, maybe the diagram is such that \(\angle JKM\) is the exterior angle, and the two interior angles that add up to it are \(\angle MKL\) and \(\angle MLK\)? No, that doesn't fit with the triangle angle sum. Wait, let's re - express the Exterior Angle Theorem correctly. The exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. So for exterior angle \(\angle JKM\), the adjacent interior angle is \(\angle MKL\), so the two remote interior angles are \(\angle KML\) and \(\angle MLK\). So \(m\angle JKM=m\angle KML + m\angle MLK\), which is the second option. Wait, but let's check the other options:

- Option 1: \(m\angle MKL + m\angle MLK=m\angle JKM\). As we saw in the example, \(50 + 60=110\), \(\angle JKM = 130\), so this is false.
- Option 2: \(m\angle KML + m\angle MLK=m\angle JKM\). In the example, \(70+60 = 130=\angle JKM\), so this is true.
- Option 3: \(m\angle MKL + m\angle MLK = 180^{\circ}\). In a triangle, the sum of two interior angles is less than \(180^{\circ}\) (since the sum of all three is \(180^{\circ}\)), so this is false.
- Option 4: \(m\angle JKM + m\angle MLK=180^{\circ}\). From the example, \(130 + 60=190 eq180\), so this is false.

Answer:

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