Question

fencing x-braces are also used to provide support in rectangular fencing. if $ab = 6$ feet, $ad = 2$ feet, and $m\angle dae = 65^\circ$, find $m\angle edc$. round to the nearest tenth, if necessary. $m\angle edc = \square$

Explanation:

Step1: Identify right triangle ADE

In rectangle \(ABCD\), \(\angle ADE = 90^\circ\), so \(\triangle ADE\) is right-angled.

Step2: Calculate \(\angle AED\)

Sum of angles in a triangle is \(180^\circ\).
\(\angle AED = 180^\circ - 90^\circ - 65^\circ = 25^\circ\)

Step3: Recognize vertical angles

\(\angle AED\) and \(\angle CEB\) are vertical angles, so \(\angle CEB = 25^\circ\).

Step4: Analyze \(\triangle EDC\) properties

In rectangle \(ABCD\), \(AB = DC = 6\) ft, \(AD = BC = 2\) ft. The diagonals of a rectangle are equal and bisect each other, so \(DE = EC\), making \(\triangle EDC\) isosceles with \(\angle EDC = \angle ECD\).

Step5: Find \(\angle DEC\)

\(\angle AED\) and \(\angle DEC\) are supplementary.
\(\angle DEC = 180^\circ - 25^\circ = 155^\circ\)

Step6: Calculate \(\angle EDC\)

Sum of angles in \(\triangle EDC\) is \(180^\circ\).
\(\angle EDC = \frac{180^\circ - 155^\circ}{2} = 12.5^\circ\)

Answer:

\(12.5^\circ\)