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Answer:
Corresponding angle theorem, vertical angle theorem, and the transitive property of congruence.
Step-by-step explanation:
Considering a set of parallel lines cut by a transversal. (Refer the attached image).
Now that lines J and K are parallel, then by "corresponding angle theorem"
By "vertical angle theorem"
Using, "transitive property of congruence"
And that is our required proof. In this whole proof we have used "corresponding angle theorem", "vertical angle theorem", and the "transitive property of congruence".
The given pentagon can be divided into two figures : a triangle and rectangle as shown in figure.
Let us find area of each figure separately.
Triangle:
Area of triangle is given by:
where b=base and h=height
Height of triangle :
h=(3x+5)-(2x+1)
h=x+4
Base = 2x-2
Area of triangle= (x-1)(x+4) = x²+3x-4
Area of rectangle:
Area of rectangle is given by:
A=l*b
Area of rectangle = 4x²-2x-2
Area of pentagon = Area of triangle + Area of rectangle
Area of pentagon = x²+3x-4 + 4x²-2x-2
Area of pentagon = 5x²+x-6
If area of pentagon is 42 cm², then solving for x,
Factorising to get x,
x=-3.2 and x=3
If we take x as negative the side will be negative, so we neglect x=-3.2
So x=3 is the answer.