# Question Video: Finding a Side Length and an Angle Using Parallel Lines and Traversals Mathematics

Consider triangle 𝐴𝐵𝐶 and lines 𝐴𝑀 and 𝐸𝐷, which are parallel to line 𝐶𝐵. Find the length of the line segment 𝐴𝐵. Find the measure of ∠𝐴𝐵𝐶.

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### Video Transcript

Consider triangle 𝐴𝐵𝐶 and lines 𝐴𝑀 and 𝐸𝐷, which are parallel to line 𝐶𝐵. Find the length of the line segment 𝐴𝐵. Find the measure of angle 𝐴𝐵𝐶.

We are given three parallel lines and two transversals of these lines. We can then recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Since 𝐴𝐸 is equal to 𝐸𝐶, the segments of the other transversal must be equal in length. So 𝐴𝐷 is equal to 𝐷𝐵, which equals five millimeters. Since 𝐴𝐵 is equal to 𝐴𝐷 plus 𝐷𝐵, then 𝐴𝐵 equals five millimeters plus five millimeters, which equals 10 millimeters. 𝐴𝐵 is equal to 10 millimeters.

It appears in the diagram that triangle 𝐴𝐵𝐶 is a right triangle. However, we need to justify why this is the case. We can do this by recalling that if a line is perpendicular to line 𝐿, then it is perpendicular to any line parallel to 𝐿. Since line 𝐸𝐷 is perpendicular to line 𝐴𝐶 and line 𝐸𝐷 is parallel to line 𝐵𝐶, we must have that lines 𝐵𝐶 and 𝐴𝐶 are perpendicular. This means the angle at 𝐶 has a measure of 90 degrees, so 𝐴𝐵𝐶 is a right triangle.

The sum of the measures of the interior angles in a triangle is 180 degrees. So 180 degrees equals 35 degrees plus 90 degrees plus the measure of angle 𝐴𝐵𝐶. Rearranging the equation, we have the measure of angle 𝐴𝐵𝐶 equals 180 degrees minus 35 degrees minus 90 degrees, which equals 55 degrees. The answers to the two parts of this question are 10 millimeters and 55 degrees.