Chapter 8 of NCERT named "Quadrilaterals" of class IX
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CLASS-IX Chapter- QUADRILATERALS
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Angle Sum Property Of 01 Quadrilateral What is the sum of angles in triangle ADC? We know that angle DAC+ angle ACD+ angle D = 180 Similarly in triangle ABC, angle CAB + angle ACB + angle B = 180 Adding 1 and 2 we get , angles DAC + + D + CAB + + B -180 + 180'= 360 Also, angles DAC + CAB = angle A and angle ACD + angle ACB = angle C So, angle A + angle D +angle B + angle C = 360' i.e. THE SUM OF THE ANGLES OF A QUADRILATERAL IS 360. ' C B
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C B Cut out a parallelogram from a sheet of paper and cut it along a diagonal. Now we obtain two triangles. What can we say about these triangles? Place one triangle over the other. Turn one round. We observe that the two triangles are congruent to each other. Thus we can say that — A diagonal of a parallelogram divides it into two congruent triangles.
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A diagonal of a parallelogram divides it into two congruent triangles. In a parallelogram ,opposite sides are equal. If each pair of opposite sides of quadrilateral is equal then it is a parallelogram. In a parallelogram opposite angles are equal. If in a quadrilateral each pair of opposite angles is equal then it is a parallelogram. The diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other then it is a parallelogram.
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A Special Condition For a 011 Il ill We have studied many properties of a parallelogram in this chapter and we have also verified that if in a quadrilateral any one of those properties is satisfied, then it becomes a parallelogram. There is yet another condition for a quadrilateral to be a parallelogram. It is stated as follows: O A QUDRILATERAL IS A PARALLELOGRAM IF A PAIR OF OPPOSITE SIDES IS EQUAL AND PARALLEL.
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Erample L) apply the jpeciol condition C Example: ABCD is a parallelogram in which P and Q are mid A B p points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that: 1. APCQ is a parallelogram 2. DPBQ is a parallelogram 3. PSQR is a parallelogram SOLUTION: I . In quadrilateral APCQ, AP is parallel to QC AP = , CQ CD , AB = CD, AP = CQ Therefore APCQ is a parallelogram. (theorem 8.8) 2.Similarly quadrilateral DPBQ is a parallelogram because DQ is parallel to PB and DQ = PB 3. In quadrilateral PSQR SP is parallel to QR and SQ is parallel to PR. SO ,PSQR is a parallelogram.
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E B -The Mid -Theorem ACTIVITY O Draw a triangle and mark the mid points E and F of the two sides of the triangle. Join the points E and F. F Measure EF and BC. Measure angle AEF and angle ABC. OBSERVATION- C EF = 1/2 BC And angle AEF = angle ABC so EF is parallel to BC . After performing this theorem we arrive at the following theorem . O The line segment joining the mid points of the two sides of a triangle is parallel to the third side. O The line drawn through the mid points of the two sides of a triangle is parallel to another side and bisects the third side.
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SUMMARY IMPORTANT TERMS - Sum of the angles of a quadrilateral is 360. A diagonal of a parallelogram divides it into two congruent triangles. In a parallelogram, Opposite sides are equal , opposite angles are equal , diagonals bisect each other. A line drawn through the mid point of a side of a triangle parallel to another side bisects the third side. The quadrilateral formed by joining the mid points of the sides of a quadrilateral in order is a parallelogram.
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EXERCISE Q .1- What is the sum of angles in a quadrilateral ? Q . 2- What are the conditions for a quadrilateral to be a parallelogram ? Q . 3- State the converse of mid point theorem ?
Discussion
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