Triangle K M L is shown. Line L K extends through point J to form exterior angle J K M.

Question

Which statement regarding the diagram is true?

m∠MKL + m∠MLK = m∠JKM
m∠KML + m∠MLK = m∠JKM
m∠MKL + m∠MLK = 180°
m∠JKM + m∠MLK = 180°

Answer 1

In triangle K M L, when considering the exterior angle $\mathrm{\angle }JKM$ formed by extending line $LK$ through point $J$, the following relationships hold true based on the properties of triangles and exterior angles.

The exterior angle theorem states that the measure of an exterior angle (in this case, $\mathrm{\angle }JKM$) is equal to the sum of the measures of the two remote interior angles (the angles inside the triangle that are not adjacent to the exterior angle). In triangle K M L, the two remote interior angles are $\mathrm{\angle }MKL$ and $\mathrm{\angle }KML$.

Thus, we can say:

$$m\mathrm{\angle }JKM=m\mathrm{\angle }MKL+m\mathrm{\angle }KML$$

Now let's analyze the statements you provided:

1. m∠MKL + m∠MLK = m∠JKM - This is incorrect because $\mathrm{\angle }MLK$ is not one of the two angles contributing to the exterior angle.

2. m∠KML + m∠MLK = m∠JKM - This is also incorrect because $\mathrm{\angle }MLK$ is not one of the angles that adds to the exterior angle $\mathrm{\angle }JKM$.

3. m∠MKL + m∠MLK = 180° - This is incorrect, as it is not necessarily true for the angles inside the triangle.

4. m∠JKM + m∠MLK = 180° - This is correct because the exterior angle $\mathrm{\angle }JKM$ and the adjacent interior angle $\mathrm{\angle }MLK$ (being on the same line) will sum to $180°$.

Therefore, the correct statement is:

m∠JKM + m∠MLK = 180°.