## Geometry: Common Core (15th Edition)

$HG = 2$ $EF = 8$ $CD = 5$
According to the trapezoid midsegment theorem, in a quadrilateral that is a trapezoid, the midsegment is parallel to the bases and is half the sum of the base lengths. Let's set up the equation to find the value of $x$: $x = \frac{1}{2}[(x - 3) + (2x - 2)]$ Evaluate parentheses first: $x = \frac{1}{2}(3x - 5)$ Divide both sides by $\frac{1}{2}$ to get rid of the fraction. $2(x) = 3x - 5$ Subtract $3x$ from both sides of the equation to move variable terms to the left side of the equation: $-x = -5$ Divide both sides by $-1$ to solve for $x$: $x = 5$ $HG = x - 3$ Let's substitute $5$ for $x$: $HG = 5 - 3$ Subtract to solve: $HG = 2$ Let's look at the expression for the longer base: $EF = 2x - 2$ Substitute $5$ for $x$ $EF = 2(5) - 2$ Multiply first, according to order of operations: $EF = 10 - 2$ Subtract to solve: $EF = 8$ Finally, let's look at the expression for the midsegment: $CD = x$ Substitute $5$ for $x$: $CD = 5$