Charles D. answered 12/16/13
Patient and Knowledgeable Tutoring by Charles
First of all it helps to think of a rectangle as a shape with 4 sides, 2 sides of which are not necessarily the same length as the other two sides. The four angles of a rectangle have a measure of 90 degrees each. There are 2 diagonals in each rectangle which have equal lengths and can split the 90 degree corner angles into smaller acute angles. Now draw a rectangle on some paper. Label the corners, starting with the lower left corner and going clockwise around the rectangle E, I, F, and G. Now draw a diagonal from corner I to corner G and another diagonal from corner E to corner F. The side EI is represented by the equation 3x+8 so label side EI with the expression 3x+8. The opposite side, FG, has the same length as side EI so label side FG as 3x+8. THE problem then states that side FI is represented by the expression 5x-6 so label side FI as 5x-6. The side opposite side FI which is EG is the same length as side FI so label side EG as 5x-6. we now have found the answer for the side EG as 5x-6. We are given that m∠IGF = 29 degrees so start at corner I and follow the diagonal to corner G then up to corner F. The angle between diagonal IG and side FG is 29 degrees. The total measure of angle G is 90 degrees so subtract 29 degrees from 90 degrees to get 61 degrees, thus, the angle between diagonal IG and side EG is 61 degrees. We know that side IF is parallel to side EG since the shape is a rectangle, thus, diagonal IG is also a transversal and the opposite interior angles of the transversal are equal. This means that the angle between diagonal IG and side EI is also 29 degrees and the angle between diagonal IG and side FI is also 61 degrees. Now we have enough information to solve for the measure of angle IFE. The diagonal EF is the mirror image of diagonal IG. Angle IFE is the same measure as the angle IGE which is 61 degrees. That completes the problem. Hope this helps.