Question

4.
m∠bcd = ____ m∠ade = ____
m∠abd = ____ m∠aeb = ____
m∠cbe = ____ m∠dea = ____

Explanation:

Step1: Identify rectangle properties

In rectangle \(ABCD\), all interior angles are \(90^\circ\), and diagonals are equal and bisect each other (\(DE = BE = CE = AE\)).

Step2: Calculate \(m\angle BCD\)

\(\angle BCD\) is an interior angle of the rectangle.
\(m\angle BCD = 90^\circ\)

Step3: Calculate \(m\angle ADE\)

Given \(m\angle EDC = 6^\circ\), \(\angle ADC = 90^\circ\), so:
\(m\angle ADE = 90^\circ - 6^\circ = 84^\circ\)

Step4: Calculate \(m\angle ABD\)

Since \(AB \parallel DC\), \(\angle ABD = \angle EDC\) (alternate interior angles).
\(m\angle ABD = 6^\circ\)

Step5: Calculate \(m\angle CBE\)

First, in \(\triangle DEC\), \(DE = CE\), so \(\angle ECD = \angle EDC = 6^\circ\). Then \(\angle DBC = 90^\circ - 6^\circ = 84^\circ\). Since \(DE = BE\), \(\angle EBD = \angle EDC = 6^\circ\).
\(m\angle CBE = 84^\circ - 6^\circ = 78^\circ\)

Step6: Calculate \(m\angle DEA\)

In \(\triangle DEC\), \(\angle DEC = 180^\circ - 6^\circ - 6^\circ = 168^\circ\). \(\angle DEA\) is supplementary to \(\angle DEC\).
\(m\angle DEA = 180^\circ - 168^\circ = 12^\circ\)

Step7: Calculate \(m\angle AEB\)

\(\angle AEB\) is vertical to \(\angle DEC\), so they are equal.
\(m\angle AEB = 168^\circ\)

Answer:

\(m\angle BCD = 90^\circ\)
\(m\angle ADE = 84^\circ\)
\(m\angle ABD = 6^\circ\)
\(m\angle AEB = 168^\circ\)
\(m\angle CBE = 78^\circ\)
\(m\angle DEA = 12^\circ\)