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Grade 9 → Math → Quadrilaterals → 8.1-Properties of a Parallelogram
- Understand properties of a parallelogram.
Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent triangles.
Theorem 8.2 : In a parallelogram, opposite sides are equal.
Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Theorem 8.4 : In a parallelogram, opposite angles are equal.
Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Theorem 8.6 : The diagonals of a parallelogram bisect each other.
Theorem 8.7 : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.:
Solved Example: 8-1-01
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Check Answer
Solution:
Proof that a Parallelogram with Equal Diagonals is a Rectangle
Let ABCD be a parallelogram with diagonals AC and BD. Given that AC = BD. In a parallelogram, the diagonals bisect each other, so let O be the midpoint of both diagonals.
Hence, AO = OC and BO = OD. Since AC = BD, we have AO + OC = BO + OD.
This implies AO = BO and OC = OD. Triangles AOB and COD are congruent by SAS (side-angle-side) criterion. Therefore, angle AOB = angle COD. As opposite angles of a parallelogram are equal, each angle must be 90 degrees.
Thus, ABCD is a rectangle.
Solved Example: 8-1-02
Show that the diagonals of a square are equal and bisect each other at right angles.
Solved Example: 8-1-03
Diagonal AC of a parallelogram ABCD bisects ∠ A (see Fig. 8.11). Show that:
(i) It bisects ∠ C also,
(ii) ABCD is a rhombus.
Check Answer
Solution:
(i) From ∆DAC and ∆BCA, show ∠DAC =∠BCA and ∠ACD =∠CAB, etc.
Solved Example: 8-1-04
ABCD is a rectangle in which diagonal AC bisects ∠A as well as∠C.Show that:
(i) ABCD is a square
(ii) Diagonal BD bisects ∠ B as well as ∠ D.
Solved Example: 8-1-05
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.12). Show that:
(i) ∆ APD ≅ ∆ CQB
(ii) AP=CQ
(iii) ∆ AQB ≅ ∆ CPD
(iv) AQ=CP
(v) APCQ is a parallelogram
Solved Example: 8-1-06
ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.13). Show that:
(i) ∆ APB ≅ ∆ CQD
(ii) AP = CQ
Solved Example: 8-1-07
ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.14). Show that:
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ∆ABC ≅ ∆BAD
(iv) Diagonal AC = diagonal BD
[Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
