Question

a plot of land has been surveyed for a new housing development with borders ab, bc, dc, and da. the plot of land is a right trapezoid with a height of 60 feet and an opposite leg length of 65 feet. if the measure of angle bcd is 67.4°, the measure of angle abc is °. if the length of base ab is 80 feet, the length of dc is feet.

Explanation:

Step1: Find the measure of angle \(ABC\)

Since \(AB\parallel DC\) in trapezoid \(ABCD\), we know that \(\angle ABC + \angle BCD= 180^{\circ}\) (consecutive - interior angles of parallel lines).
Given \(\angle BCD = 67.4^{\circ}\), then \(\angle ABC=180^{\circ}-\angle BCD\)

$$ \angle ABC = 180^{\circ}- 67.4^{\circ}=112.6^{\circ} $$

Step2: Find the length of \(DC\)

Let's draw a right - triangle by dropping a perpendicular from \(C\) to \(AB\) (say \(CE\perp AB\), and \(AE = DC\) because \(ADCE\) is a rectangle).
Using the Pythagorean theorem in the right - triangle \(BCE\) (where \(BC = 65\) ft and \(CE = 60\) ft). Let \(BE\) be the base of the right - triangle. By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = BC\), \(a = CE\), and \(b = BE\)

$$ BE=\sqrt{BC^{2}-CE^{2}}=\sqrt{65^{2}-60^{2}}=\sqrt{(65 + 60)(65 - 60)}=\sqrt{125\times5}=\sqrt{625}=25 $$

Since \(AB=AE + BE\) and \(AE = DC\), then \(DC=AB - BE\)
Given \(AB = 80\) ft, so \(DC=80 - 25=55\) ft

Answer:

The measure of angle \(ABC\) is \(112.6^{\circ}\) and the length of \(DC\) is \(55\) feet.