The diagram for this problem is shown below. Point E is the intersection of the two diagonals and we know that:
[tex]\angle BCE=\beta \\ \\ \angle DCE=\theta[/tex]
Every internal angle of any rectangle measures 90 degrees, so:
[tex]\beta +\alpha=90 \\ \\ (4x-23)+(5+5x)=90 \\ \\ \\ Isolating \ x: \\ \\ (4x+5x)+(5-23)=90 \\ \\ 9x-18=90 \\ \\ 9x=108 \\ \\ x=\frac{108}{9}=12[/tex]
So:
[tex]\beta=4x-23=4(12)-23=25^{\circ}[/tex]
So the measure of angle BEC can be found as follows:
We know that triangle ΔCEB is an isosceles triangle because the diagonals of any rectangle measure the same. So
[tex]\angle EBC=\beta[/tex]
The sum of internal angles of any triangle add up to 180 degrees, so:
[tex]\beta + \beta+\angle BEC=180^{\circ} \\ \\ 2\beta+\angle BEC=180^{\circ} \\ \\ \angle BEC=180^{\circ}-2\beta \\ \\ \angle BEC=180^{\circ}-2(25^{\circ}) \\ \\ \angle BEC=180^{\circ}-50^{\circ} \\ \\ \boxed{\angle BEC=130^{\circ}}}[/tex]
The sum of the angles of BCE and DCE is 90 degrees. The value of x is 12 degrees. Then the measure of angle ∠BEC is 130 degrees.
It is a polygon that has four sides. The sum of the internal angle is 360 degrees. In a rectangle, opposite sides are parallel and equal and each angle is 90 degrees. And its diagonals are also equal and intersect at mid-point.
ABCD is a rectangle.
The measure of angle BCE is 4x - 23 and the measure of angle DCE is 5 + 5x.
We know that the sum of the angles of BCE and DCE is 90 degrees. Then we have
∠BCE + ∠DCE = 90
4x - 23 + 5 + 5x = 90
9x - 18 = 90
9x = 108
x = 12
Then the angle ∠BEC will be
∠BEC = 180 - 2∠BCE
∠BEC = 180 - 2 [4(12) - 23]
∠BEC = 180 - 50
∠BEC = 130°
More about the rectangle link is given below.
https://brainly.com/question/10046743
