Exercise 8.1 Part 2
Question 5: Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Answer: Using the same figure,
(Angles opposite to equal sides are equal)
So, all angles of the quadrilateral are right angles making it a square.
Question 6: Diagonal AC of a parallelogram ABCD bisects angle A . Show that
(i) it bisects angle C also,
(ii) ABCD is a rhombus.
Answer: ABCD is a parallelogram where diagonal AC bisects angle DAB
In ΔADC and ΔABC
`∠DAC=∠`BAC (diagonal is bisecting the angle)
`AC=AC` (common side)
`AD=BC` (parallel sides are equal in a parallelogram)
This proves that AC bisects ∠DCB as well.
Now let us assume another diagonal DB intersecting AC on O.
As it is a parallelogram so DB will bisect AC and vice-versa.
In ΔAOD and ΔBOD
(opposite angles are equal in parallelogram so their halves will be equal)
Hence `ΔAOD ≅ ΔBOD`
As diagonals are intersecting at right angles so it is a rhombus
Question 7: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that:
- `ΔAPD≅ ΔCQB`
- APCQ is a parallelogram
Answer: In ΔAPD and ΔCQB
`AD=BC` (opposite sides are equal)
`∠DAP=∠BCQ` (opposite angles’ halves are equal)
So, `AP=CQ` proved
In ΔAQB and ΔCPD
`AB=CD` (opposite sides are equal)
`∠BAQ=∠DCP` (opposite angles’ halves are equal)
So, `AQ=CP` proved
(corresponding angles of congruent triangles APD and CQB)
In ΔDQP and ΔBQP
(from previous proof)
`PQ=PQ` (common side)
With equal opposite angles and equal opposite sides it is proved that APCQ is a parallelogram