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Quadrilateral. 45 0. 2 pairs of equal adjacent sides. 1 pair of // opp. Sides. Kite. Trapezium. Sum of interior angles is 180 0. One of the diagionals is axis of symmetry. Parallelogram. 2 diagionals are . 2 pairs of opp.// sides. 4 equal sides. Rhombus. Properties of trapesium.

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1. Quadrilateral 450 2 pairs of equal adjacent sides 1 pair of // opp. Sides Kite Trapezium Sum of interior angles is 1800 One of the diagionals is axis of symmetry Parallelogram 2 diagionals are  2 pairs of opp.// sides 4 equal sides Rhombus Properties of trapesium 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal (opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Properties of // gram and kite Diagonals bisects each interior angle 4 right angles Rectangle 4 right angles and 4 equal sides Properties of // gram Properties of rhombus/rectangle Diagonals are equal Angles between each diagional and each side is 450 Square

2. Trapeziums Sum of interior angles is 1800 Definition : 1 pair of parallel sides Properties:

3. Parallelogram 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Definition : 2 pairs of opp. parallel sides Properties:

4. Conditions for Parallelogram If 2 pairs of opposite sides are equal thenthe quadrilateral is parallelogram.(opp. sides eq.) If 1 pair of opposite sides is equal and parallel thenthe quadrilateral is parallelogram (opp. sides eq. and //) If diagonals bisect each other thenthe quadrilateral is parallelogram (diag. Bisect each other) If 2 pairs of opposite angles are equal then the quadrilateral is parallelogram.(opp. s of eq.)

5. Rhombus Definition : a // gram or a kite of 4 equal sides 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisects each interior angle Diagonals are  Diagonals bisect each other (diag. Of // gram) Properties:

6. Rectangle 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Diagonals are equal Definition : a parallelogram of 4 right angles Properties:

7. Square Definition : a // gram of 4 right angles and 4 equal sides 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) 450 Diagonals are equal Diagonals are  Angles between each diagonal and each side is 450 Properties: 2 pairs of opposite sides are equal.(opp. sides of // gram)

8. Example 1: In the figure, PQRS is a kite P y+3 x+1 S Q x+y 8 R • Find x and y. • Find the perimeter of the kite PQRS PQ = PS (given) x+1 = y+3x-y=2 (1) QR=SR (given)x+y=8 (2) (1)+(2), 2x=10 x=5Put x=5 into (1), 5-y=2 y=3 (a) (b) PQ = x+1=5+1=6 PQ+PS+SR+QR = 6 + 6 + 8 + 8 =28

9. Example 2: In the figure, ABCD is a kite. E is a point of intersection of diagonals AC and BD, AE=9 cm, EC=16 cm and DE=EB=12 cm D 12 16 C A 9 E 12 B • Find the area of ABCD. • Find the perimeter of ABCD • ABC= ADC (axis of symmetry AC)AED=900 Area of ADC = Area of kite ABCD=Area of ABC+Area of ADC = 150+150 =300 cm2 In ADE, AD2=AE2+DE2=92+122=225 cm2 (Pyth theorem) AD=15 cmIn CDE, DC2=DE2+EC2=122+162=400 cm2 (Pyth theorem) DC=20 cm  Perimeter of ABCD=AD+AB+ DC+CB = 15 + 15 + 20 + 20 =70 cm (b)

10. Example 3: In the figure, ABCD is a parallelogram. Find x and y. A D x 1500-y 2y 680 B C AD//BC (Given) x+680=1800(prop. Of trapezium)  x=1120(1500-y)+2y=1800(prop. Of trapezium) 1500+y=1800  y=1800 -1500=300

11. Example 4: In the figure, ABCD is a parallelogram. Find x and y. D C 3x+100 x+200 y A B DAB=DCB (opp. s of // gram) x+200=3x-100 2x=300x=150DAB+CBA=1800(int.s , AD//BC) x+200+y=1800 150+200+y=1800y=1450

12. Example 5: In the figure, ABCD is a isosceles trapezium with AB=DC.Find x , y and z A D z 1260 a Construct AE // DC  AD//EC and AE//DC ADCE is a parallelogram (Definition of // gram) E x y  B C  ADCE is a parallelogram (proof)AE=DC (opp.sides of // gram)  AD//BC (Given) x+1260=1800(prop. Of trapezium)  x=540 In ABE,AE=DC (proof) AB=AC (given) AB=AE  y=a (base s. isos )a= x(corr.s. AE//DC)  y=x =540 y+z=1800(prop. Of trapesium)z= 1800-540 = 1260

13. MID-POINT THEOREM A N M C B IF AM = MB and AN =NC then (a) MN // BC (b) MN = (Abbreviation: Mid-point theorem)

14. Example 13: In the figure, ABC is a triangle, find x and y. C y E 6 420 x B A D CE=BE (given) AD=DB (given) (mid-point theorem) DE//AC x = EDB =420 (corr. s , DE//AC) (mid-point theorem)

15. Example 14: Prove that BPQR is a parallelgram A R Q C B P (given) AR=RB (given) AQ=QC (mid-point theorem) (given) (opp-sides eq. And //)

16. Ex 11D D A M y cm x cm 5 cm B N C 1(b) BM=MD (given) BN=NC (given) AM=AC (given) BN=NC (given) (mid-point theorem) (mid-point theorem)

17. Ex 11D A 1100 a Q P 460 C B 2(b) AP=BP (given) AQ=CQ (given) (mid-point theorem) (corr.s. PQ//BC) In APQ, APQ+ PAQ+ a = 1800 460+1100+a=1800 a=240 (adj s. on a st line)

18. A 10 8 F E A B C D 9 60 F E 70 50 3(a) 3(b) B C D 9

19. 4. BP=PA (given) CR=RB (given) C (mid-point theorem) P R AQ=QB (given) AP=PC (given) B A Q Area of ABC  (mid-point theorem) 6 8

20. INTERCEPT THEOREM transversal X intercept Y B P A D C Q

21. INTERCEPT THEOREM If AB//CD//EF then (intercept theorem) A B D C F E

22. INTERCEPT THEOREM Construct GB through A such that BG//CD//EF (given) GB//CD//EF B G (intercept theorem) Proved: A D C E F

23. Example 15. AP//BQ//CR, AB=BC, AP=11 and CR=5. Find BQ. Join AR to cut BQ at S (given) AP//BQ//CR (given) S (intercept theorem) (given) (proved) (proved) (proved) (mid-pt theorem) (mid-pt theorem) A B C 11 5 P Q R BQ=BS+SQ = 2.5+5.5=8

24. Example 16. AB and DC are straight lined. Find x and y. (b) AB=6, PB=2 and AQ=9. Find QC (proved) (a) Proved: A E D Join DE through A and // BC DE//PQ//BC (given) Q P (intercept theorem) B C

25. Example 16. Find QR and CD. AP//BQ//CR (given) BQ//CR//DS (given) (intercept theorem) (intercept theorem) A P 3 2 Q B 6 R C 8 S D

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