Question

if quadrilateral abcd is a square and dc = 20, find the missing values. bd = 20 m∠edc = 45° ab = 90° m∠dab = 10√2 ae = 20√2

Explanation:

Step1: Recall square properties

In a square, all sides are equal. So if \(DC = 20\), then \(AB=20\) since \(AB = DC\) in square \(ABCD\).

Step2: Use Pythagorean theorem for diagonal

In right - triangle \(BCD\) with \(BC = DC = 20\), by the Pythagorean theorem \(BD=\sqrt{BC^{2}+DC^{2}}=\sqrt{20^{2}+20^{2}}=\sqrt{400 + 400}=\sqrt{800}=20\sqrt{2}\).

Step3: Find angle measure

The diagonals of a square bisect the angles of the square. In square \(ABCD\), \(\angle ADC=90^{\circ}\), and the diagonal \(AC\) bisects \(\angle ADC\), so \(m\angle EDC = 45^{\circ}\). Also, each interior angle of a square is \(90^{\circ}\), so \(m\angle DAB=90^{\circ}\).

Step4: Diagonal bisection

The diagonals of a square bisect each other. Since \(BD = 20\sqrt{2}\), then \(AE=\frac{1}{2}AC\) (and \(AC = BD\)), so \(AE = 10\sqrt{2}\).

Answer:

\(AB = 20\), \(BD=20\sqrt{2}\), \(m\angle EDC = 45^{\circ}\), \(m\angle DAB = 90^{\circ}\), \(AE = 10\sqrt{2}\)