In the given figure,QR//PB,parallelogram PQRS and AQRB are standing on the same base QR.prove that,□PQRS=□AQRB
i)this figures
ii)♢PAQ=♢SBR
iii)□PQRB-♢PAQ=□PQRB-♢SBR
iv)AQRB=□PQRS
To prove that quadrilateral PQRS is equal to quadrilateral AQRB, we need to show that they have the same area.
Given:
1. QR is parallel to PB.
2. Quadrilateral PQRS is a parallelogram.
3. Both PQRS and AQRB are standing on the same base QR.
Proof:
Since QR is parallel to PB, we can use the property of parallel lines to establish the following:
∠QRB = ∠PBR (Corresponding angles) (1)
∠QRB + ∠PBR = 180° (Angles on a straight line) (2)
Since PQRS is a parallelogram, opposite angles are equal:
∠PRS = ∠PQR (3)
∠QRS = ∠RPQ (4)
Consider the triangles QRB and PBR. They share an equal base QR, and from equation (1) we know that:
∠QRB = ∠PBR (5)
Using the Angle-Angle similarity criterion, we can conclude that triangles QRB and PBR are similar.
Similarly, the triangles PRS and PQR are similar because:
∠PRS = ∠PQR (from equation (3))
Since the triangles QRB and PBR are similar, their corresponding sides are proportional:
⇒ QR/PB = QB/PR (6)
Since the triangles PRS and PQR are similar, their corresponding sides are also proportional:
⇒ QR/PB = PR/PQ (7)
From equations (6) and (7), we can conclude that:
QB/PR = PR/PQ
⇒ QB * PQ = PR²
Now, let's consider the quadrilaterals PQRS and AQRB:
We can divide quadrilateral PQRS into two triangles, PQR and PRS.
Similarly, we can divide quadrilateral AQRB into two triangles, AQB and QRB.
As we have proved, the triangles QRB and PBR are similar. Therefore, their areas are in proportion to the squares of their corresponding sides:
Area of QRB = (1/2) * QB * PR
Area of PBR = (1/2) * PQ * PR (from equation (7))
Since QB * PQ = PR² (from our earlier observation), we can conclude that the area of QRB is equal to the area of PBR.
Similarly, we can show that the areas of triangles PQR and AQB are equal.
Now, adding the areas of triangles PQR and PRS, we get the area of quadrilateral PQRS.
The area of quadrilateral PQRS = Area of triangle PQR + Area of triangle PRS
= (1/2) * PQ * PR + (1/2) * PQ * PR
= (1/2) * PQ * PR + (1/2) * PQ * PR
= PQ * PR
Adding the areas of triangles AQB and QRB, we get the area of quadrilateral AQRB.
The area of quadrilateral AQRB = Area of triangle AQB + Area of triangle QRB
= (1/2) * QB * PR + (1/2) * QB * PR
= (1/2) * QB * PR + (1/2) * QB * PR
= QB * PR
Since PQ * PR = QB * PR, we can conclude that the area of quadrilateral PQRS is equal to the area of quadrilateral AQRB:
□PQRS = □AQRB
Thus, we have proved that the two quadrilaterals have the same area.