Parallelogram and its Theorems
In this section we will discuss parallelogram and its theorems.On the basis of properties of parallelogram there are different theorems.
Examples
1) A diagonal of a parallelogram divides it into two congruent triangles.
Given : Δ ABCD is a parallelogram and AC is a diagonal.
Prove that : ΔABC = Δ CDA
Statements | Reasons |
ABCD is a parallelogram | Given |
AC is a diagonal given | Given |
∠BCA = ∠DAC | Alternate interior angles |
∠BAC = ∠DCA | Alternate interior angles |
AC = CA | Reflexive (common side) |
ΔABC = ΔCDA | ASA postulate |
So diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.
Statements of parallelogram and its theorems 1) In a parallelogram, opposite sides are equal. 2) If each pair of opposite sides of a quadrilateral is equal then it is a parallelogram. 3) In a parallelogram, opposite angles are equal. 4) If in a quadrilateral, each pair of opposite angles is equal then it is a parallelogram. 5) The diagonals of a parallelogram bisect each other. 6) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. 7) A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. |
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Some solved examples using parallelogram and its theorems
1) Two opposite angles of a parallelogram are ( 3x – 2) ^{0} and
(50 – x ) ^{0} .
Find the measure of each angle of the parallelogram.
Solution :
Opposite angles of parallelogram are equal.
3x – 2 = 50 – x
⇒ 3x + x = 50 + 2
⇒ 4x = 52
∴ x = 13
1st angle = 3x – 2 = 3(13) – 2 = 37 ^{0}
2nd angle = 37 ^{0}
3rd angle = 180- 37 = 143 ^{0}
4th angle = 143 ^{0}
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2) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ.
Given : ABCD is a parallelogram. DP = BQ.
Prove that 1) ΔAPD = ΔCQB
2) AP = CQ 3) Δ AQB = ΔCPD 4) AQ =CP 5) APCQ is a parallelogram.
Statements | Reasons |
1) ABCD is a parallelogram | 1) Given |
2) DP = BQ | 2) Given |
3) AD = BC | 3) Properties of parallelogram |
4) ∠ADP = ∠CBQ | 4) Alternate interior angles |
5) ΔAPD =ΔCQB | 5) SAS Postulate |
6) AP = CQ | 6) CPCTC |
7) AB = CD | 7) Properties of parallelogram |
8) BQ = DP | 8) Given |
9) ∠ABQ = ∠CDP | 9) Alternate interior angles |
10) ΔAQB = ΔCPD | 10) SAS Postulate |
11) AQ = PC | 11) CPCTC |
12) APCQ is a parallelogram | 12) Two pairs of opposite sides of ≅then it is parallelogram. From (6) and (11) |
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3) ABCD is a parallelogram and AP and CQ are perpendiculars from A and C on diagonal BD.Prove that AP = CQ.
Statements | Reasons |
1) ABCD is a parallelogram | 1) Given |
2) AP and CQ are perpendiculars | 2) Given |
3) ∠APB = ∠CQD | 3) Definition of perpendiculars.Each of measure 90^{0} |
4) ∠ABP = ∠CDQ | 4) Alternate interior angles |
5) AB = CD | 5) Properties of parallelogram |
6) ΔAPB = ΔCQD | 6) AAS |
7) AP = CQ | 7) CPCTC |
Quadrilateral
• Introduction to Quadrilateral
• Types of Quadrilateral
• Properties of Quadrilateral
• Parallelogram and its Theorems
• Rectangle and its Theorems
• Square and its Theorems
• Rhombus and its Theorems
• Trapezoid (Trapezium)and its Theorems
• Kite and its Theorems
• Mid Point Theorem