Answer:
x = 1
Explanation:
A rectangle has two diagonals that are equal in length and bisect each other. That means they divide each other into two equal parts. So, if E is the intersection of the diagonals of rectangle ABCD, then AE = EC and BE = ED.
Also, since a diagonal divides a rectangle into two right triangles, we can use the fact that the sum of angles in a triangle is 180 degrees to find x.
Let’s look at triangle AEB first. The measure of angle AEB is 13x, so we can write:
13x + angle ABE + angle EBA = 180
We don’t know angle ABE or angle EBA yet, but we can use the fact that opposite angles of a parallelogram are equal to find them. Since ABCD is a parallelogram (a rectangle is a special case of a parallelogram), we have:
angle ABE = angle CDE angle EBA = angle DCA
Now we can substitute these values into our equation:
13x + angle CDE + angle DCA = 180
Next, let’s look at triangle CED. The measure of angle ECD is 3x-5, so we can write:
3x - 5 + angle CDE + angle DEC = 180
We don’t know angle DEC yet, but we can use the fact that adjacent angles of a parallelogram are supplementary to find it. Since ABCD is a parallelogram, we have:
angle DEC + angle DAB = 180 angle DEC = 180 - angle DAB
Now we can substitute this value into our equation:
3x - 5 + angle CDE + (180 - angle DAB) = 180
Simplifying both equations by subtracting 180 from both sides, we get:
13x + angle CDE + angle DCA = 0 3x - 5 + angle CDE - angle DAB = 0
Now we have two equations with three unknowns: x, angle CDE and angle DAB. To solve for x, we need to eliminate one of the unknown angles. We can do this by adding or subtracting the two equations.
Let’s try adding them first:
(13x + angle CDE + angle DCA) + (3x - 5 + angle CDE -angle DAB) = 0 16x -5 +2(angle CDE) =0
This gives us an equation with only x and one unknown:
16x -5+2(angleCDE)=0
To solve for x, we need to find out what (angleCDE)is.
We can do this by using another fact about rectangles: The diagonals of a rectangle are perpendicular to each other. That means they form four right angles at their intersection point E.
So,
angle AEB+angleBEC=90 13x+angleBEC=90 angleBEC=90-13x
Similarly,
angleCED+angleDEC=90 3x-5+angleDEC=90 angleDEC=95-3x
Since opposite angles of a parallelogram are equal,
angleBEC=angleDEC 90-13x=95-3x 10=10x X=1
Therefore,
the constantofproportionalityis 1.
The correct answer is X=1.