1. The diagram shows triangle LMK with point N on side MK and point P on side LK.
2. Angle LMN is given as 50°, and angle PNK is also given as 50°.
3. Since angle LMN and angle PNK are both 50° and are corresponding angles formed by the transversal MK intersecting lines LM and PN, their equality implies that line segment LM is parallel to line segment PN (LM || PN).
4. Because LM || PN, triangle LMK is similar to triangle PNK by the Angle-Angle (AA) similarity criterion (they share angle K, and have equal corresponding angles at M and N).
5. In similar triangles, corresponding sides are proportional. Therefore, the ratio of side LM to side PN equals the ratio of side MK to side NK.
6. Side LM is labeled 'a', side PN is labeled 'x', side MK is labeled 'b + c' (since it is composed of segments MN = b and NK = c), and side NK is labeled 'c'.
7. Setting up the proportion: a / x = (b + c) / c.
8. Solving for x: x = a * c / (b + c).
Parent Tip: Review the logic above to help your child master the concept of x is given.
