Question

quadrilateral abcd is a square. find the missing values. ad = m∠cba = ac = m∠ceb = ec = m∠bac =

Explanation:

Step1: Recall square - side property

In a square, all sides are equal. Given \(AB = 18\), so \(AD=18\).

Step2: Recall square - angle property

Each interior angle of a square is \(90^{\circ}\), so \(m\angle CBA = 90^{\circ}\).

Step3: Use Pythagorean theorem for diagonal

In right - triangle \(ABC\) with \(AB = BC=18\), by the Pythagorean theorem \(AC=\sqrt{AB^{2}+BC^{2}}=\sqrt{18^{2}+18^{2}}=\sqrt{2\times18^{2}} = 18\sqrt{2}\).

Step4: Recall diagonal - intersection property

The diagonals of a square are perpendicular to each other, so \(m\angle CEB = 90^{\circ}\).

Step5: Recall diagonal - length property

The diagonals of a square bisect each other. Since \(AC = 18\sqrt{2}\), then \(EC=\frac{AC}{2}=\frac{18\sqrt{2}}{2}=9\sqrt{2}\).

Step6: Recall diagonal - angle property

The diagonals of a square bisect the angles of the square. So \(m\angle BAC = 45^{\circ}\).

Answer:

\(AD = 18\)
\(m\angle CBA=90^{\circ}\)
\(AC = 18\sqrt{2}\)
\(m\angle CEB = 90^{\circ}\)
\(EC = 9\sqrt{2}\)
\(m\angle BAC = 45^{\circ}\)