L and m are two parallel tangents at A and B. The tangent at C makes an intercept DE between L and m. Prove that angle DEF = 90 degree.
To prove that angle DEF is 90 degrees, we will need to utilize some properties of tangents and parallel lines.
Let's start by constructing the diagram based on the given information. Draw two parallel lines, L and m. Draw two tangents, one at point A and another at point B, that intersect lines L and m at points D and E, respectively. Let the point of intersection of the tangents be C.
Now, to prove that angle DEF is 90 degrees, we can use the fact that the tangent and the radius of a circle are always perpendicular to each other. So, we need to show that segment CD is perpendicular to line AB.
To do this, we can use the property of alternate angles formed by a transversal intersecting parallel lines. In this case, line AB is the transversal, and lines L and m are parallel.
Since DE is an intercept between lines L and m, we know that angle ACD and angle BCD are alternate angles, and they are congruent. This is because alternate angles are equal when a transversal intersects two parallel lines.
Similarly, angles CDA and CDB are alternate angles formed by the transversal AB and the parallel lines L and m. Hence, these angles are also congruent.
Now, we have two pairs of congruent angles: angle ACD ≅ angle BCD, and angle CDA ≅ angle CDB.
Using the properties of a tangent and the radius of a circle, we know that angle ACD is 90 degrees (because AC is the tangent to the circle at point A). Similarly, angle CDB is 90 degrees (because BC is the tangent to the circle at point B).
Since angles ACD and CDB are both 90 degrees, their sum is 180 degrees. Therefore, angles ACD and CDB are supplementary angles.
Since angle ACD and angle BCD are congruent, and they are both supplementary to angle CDB, they must each be 90 degrees.
So, we have angle ACD = 90 degrees and angle BCD = 90 degrees. Therefore, segment CD is perpendicular to line AB.
Finally, segment DE lies on line CD, which is perpendicular to line AB. Hence, angle DEF, which is formed by line DE and line AB, is a right angle or 90 degrees.
Therefore, we have proven that angle DEF = 90 degrees.