中三數學教學筆記

Chapter 08 Qudrilaterals (四邊形的特性) New Century Mathematics (Oxford Canotta Maths)

Section Topic Teaching Notes Classwork or Homework

8.1 Basic Knowledge of Quadrilaterals - Students should be able to give examples of common quadrilaterals (四邊
形).
- Students should be able to state the definition and properties of
1. kite (鷂形),
2. trapezium (梯形). Ex.8A Q.1
3. parallelogram (平行四邊形). Ex.8A Q.2
4. rectangle (矩形). Ex.8B Q.4
5. square (正方形). Ex.8A Q.5
6. rhombus (菱形). Ex.8A Q.2
8.2 Parallelograms - Students should be able to use the properties of parallelograms to solve Ex.8B Q.1-8
problems.
- Students should be able to state the conditions for proving parallelogram. Ex.8B Q.14-15
8.3 Rhombuses, Rectangles and Squares - Students should be able to use the properties to solve problems. Ex.8C Q.1-3
- Students should be able to proof various quadrilaterals. Ex.8C Q.11-12
8.4 Proofs Related to Parallelograms - Students should be able to use properties of parallelogram and put forward Ex.8D 7-8
to proof other geometric problems.
8.5 Midpoint Theorem and Intercept - Students should be able to use Midpoint Theorem (中點定理) to solve Ex.8E Q.1-12
Theorem problems.
- Students should be able to use Intercept Theorem (截線定理) to solve
problems.
Supp Some Important Geometrical - Students should be able to recall and use reasons for calculations: Handout 10-1 & 10-2
Theorem Theorems of Equal Ratios 等比定理
Converse of Equal Ratios 等比逆定理
Perpendicular Bisector Theorem 垂直平分線定理
Angle Bisector Theorem 角平分線定理
Summary of Properties of all Types of Parallelogram

Properties of Parallelogram Properties of Rectangle (Equiangular Quadrilateral) Properties of Squares (Regular Quadrilateral)
(1)-(8) properties of parallelogram
1. 2 pairs of parallel lines (definition) (1)-(8) properties of parallelogram
9. all angles are right angles (definition)
2. 2 pairs of lines of same lengths (9)-(12) properties of rectangle
10. diagonals are of the same length
3. a pair equal and parallel lines 11. sides and diagonal obey Pyth. Th. (9)-(13) properties of rhombus
12. angle betw. diag = 2 times that with side
4. 2 pairs opposite angles of same sizes 18. angle bet. side and diagonal is 45o
Properties of Rhombus (Equilateral Quadrilateral)
5. adjacent angles supplementary 19. diagonal is 2 times of side length
(1)-(8) properties of parallelogram
6. diagonals bisect each other 9. all side lengths are the same (definition) 20. area = square of side length
10. diagonal bisect interior angles
7. area = ab sin θ 1
11. diagonals perpendicular to each other 21. area = 2 × (square of diagonal)
1
8. area = 2 × (product of diagonals) × sin φ θ
12. lengths of diagonals are 2l sin 2 and

θ
2l cos 2

1
13. area = 2 × (product of lengths of diag.)
Summary of Properties of some special Quadrilateral

Properties of Kite Properties of Trapezium

1. form from 2 isosceles triangles (definition) 1. only has 1 pair of parallel lines (upper and lower bases) (definition)
2. diagonals perpendicular to each other 2. interior angles supplementary
1 1
3. area = 2 × product of lengths of diagonals 3. area = 2 × (sum of upper and lower bases) × (height)
Supplementary Notes on Quadrilaterals

1. ABCD is a square and CDE is an equilateral triangle. E

Find the size of (a) ∠ADE,
(b) ∠DAE,
(c) ∠AEB,
(d) ∠AFB, and
(e) ∠EAF.
D C

A B

D C
2. ABCD is a square and CDE is an equilateral triangle.
Find the size of (a) ∠ADE,
(b) ∠AED, and
(c) ∠AEB.
E

A B

3. ABCDEF is a regular hexagon and EFGH is a square.

Find the size of (a) ∠AFG, and F E
(b) ∠GIB, correct to 1 d.p.

A G H D

B C

4. ABCD is a parallelogram and XB bisects ∠ABC. Find θ.

D X A
θ

62o
C B
5. 下圖中 ABCD 是一個平行四邊形，求 x + y 之值。 A B
In the figure below, ABCD is a parallelogram. Find the
value of x + y. xo yo

50o
30o
D C

6. PQRS is a rhombus. If PR = 5 cm and QS = 10 cm. Find the area of the rhombus. (5 marks)
P Q

S R

7. In the figure, ABCD is a parallelogram. M is the midpoint of

CD , Y is a point on AD and AX meets BY at X. A B

If AX : XM = 1 : 2 and XY = 3, find the length of BX.

X
Y

D M C

8. In the figure, BPD is a diagonal of the rectangle ABCD. A straight

line CPQ cuts BD at P and BA at Q. If ∠CPD = 100o and
∠BDC = 30o, find the size of ∠CQA. State the reasons you used
clearly.
(5 marks)

9. In the figure, ABCD is a parallelogram. QB =

A B

Q
6

D 8 P 4 C
10. B E C
ABCD is a square of side length 10 cm.
AEDF is a rhombus. Find the perimeter of the rhombus.

A D

F
11. H G ABCDEF is a regular hexagon and EFGH is a square.
Find (a) ∠AFG,
(b) ∠FGA, and
E F (c) ∠BHE

D A

C B
Supplementary Notes on Some Important Geometric Theorems

1. In the figure, AC = CE = EF = 6 cm, and AB // CD // EF. E 6 cm F

If the length of AB = 12 cm, find the length of CD.
Show your steps and reasons clearly. 6 cm

C G D
6 cm

A 12 cm B

2. b+2
In the figure, find the value of b.
3b

12 8

3.
A B In the following figure, AB // CD // EF, AC = 3, CE = 7,
3 AB = 12 and EF = 27, find the length of CD.
C D

E F 16
A F
4. In the figure, find BE.
B E

C 10 D

5. In the figure, AC // DE, FG // BC and AD : DF : FB = 1 : 2 : 3.

If BE = 10, find FG. A
D

F G

B E C

6. A In the figure, E and F are the mid-points of AB and AD

respectively. G and H are points on BC and CD respectively
F
6 cm CG CH 3
D such that = = . If EF = 6 cm, then GH =
E GB HD 5

B
G
C
Chapter 08 Properties of Quadrilaterals Quiz 08-0
F.3_____ Name:___________________________________________________( ) Marks: _____ / 30
CONCAVE QUADRILATERAL CONVEX QUADRILATERAL 4. In the figure, the diagonals divide the quarilateral
1. Find the size of BCD. o o
2. If the exterior angles of a quadrilateral are x , 2x , ABCD into 4 parts. The areas of three parts are
o o
(3x + 40) and (4x – 10) , find the size of the shown below, find the area of the missing part.
smallest interior angle of the quadrilateral. A (3 marks)
(3 marks)

15 16
(4 marks) B E D

12

3. If the lengths of diagonals of a quadrilateral are

10 cm and 16 cm, and they make an angle 60o with

each other, find the area of the quadrilateral.

(3 marks)
 Trapezium 8. The figure shows an isosceles trapezium ABCD with 9. The figure shows a right-angled trapezium ABCD
5. In the figure, AB // upper base 8 cm and lower base 18 cm. with AB = 2 cm, BC = 3 cm and CD = 6 cm.
CD and AC = BD. If ABC = 120o, find
If CAD = 20o and (a) the height,
ADB = 80o, then  (b) the area, and
ADC = (c) the perimeter.
(6 marks)
A . 30o
B . 40o
C . 50o (a) Find the area of the trapezium ABCD.
D . 60o (b) Find the perimeter of the trapezium ABCD.
Isosceles Trapezium (c) Find the lengths of the diagonals AC and CD.
6. ABCD is a trapezium in which AB // DC, (d) Find the acute angle make between the
AB = 8 cm, DC = 18 cm, AD = BC = 13 cm. diagonals (correct to the nearest degree).
Find the area of the trapezium. (8 marks)

A. 156 cm2
B. 169 cm2
C. 216 cm2
D. 312 cm2
E. 338 cm2

Right-angled Trapezium
7. In the figure, the area of the trapezium ABCD is

A. 345 cm2
B. 349 cm2
C. 690 cm2
D. 698 cm2
Chapter 08 Properties of Quadrilaterals Quiz 08-1
F.3_____ Name:___________________________________________________( ) Marks: _____ / 18
1. The lengths of the diagonals of a kite are 12 cm and 2. A kite is left-right symmetrical. The lengths of the 3. An isosceles trapezium has side lengths 5 cm, 5 cm,
20 cm respectively. Find the area of the kite. horizontal and vertical diagonals of a kite are 30 cm 5 cm and 13 cm. Find the area of the trapezium.
(3 marks) and 56 cm respectively. If the lengths of the upper (4 marks)
2 sides are both equal to 25 cm, find the length of
either lower side of the kite. (5 marks)

25 cm
20 cm

56 cm

12 cm

30 cm
4. The lengths of the upper base, lower base and the 5. In the figure, the area of the trapezium is 96 cm2. 7. The diagonals of a quadrilateral are of lengths 10 cm
height of a right-angled trapezium are 6 cm, 12 cm and Find x. and 15 cm. The angle between them is 70o. Find
8 cm respectively. Find the perimeter of the the area of the quadrilateral.
trapezium. (3 marks) A. 1 (Correct the answer to 3 significant figures)
6 cm B. 5 (3 marks)
C. 7
D . 11
8 cm

12 cm

6. In the figure, AB //
CD and AC = BD.
If CAD = 20o and
ADB = 80o, then 
ADC =

A. 30o
B. 40o
C. 50o
D. 60o
Chapter 08 Properties of Quadrilaterals Quiz 08-2
F.3_____ Name:___________________________________________________( ) Marks: _____ / 32
1. ABCD is a parallelogram and XB bisects ABC. 4. The figure shows a parallelogram ABCD with its diagonals
7. ABCD is a parallelogram. Find the values of x
Find . meeting at E. and y. (4 marks)
D X A
If AE = 3 cm and BE = 2
A. 310  cm, find the area of the
A
o
B
(3x + 10)
B. 590 parallelogram correct to
62o
C. 620 the nearest 0.1 cm2.
D. 1180 C B
yo
A. 2.3 cm 2
(5x – 30)o
2. In the figure, ABCD is a rhombus and B. 7.7 cm2
D C
ABE is a straight line. If BCE = 40o C. 9.2 cm2
and BC= CE, thenCAD = D. 18.3 cm2

A. 35o 5. ABCD is a square and BPD is a diagonal. Find the size8.of 

PQRS is a square. Find the values of a and b.
B. 40o AQC. (2 marks)
C. 45o
D. 50o A. 95o
B. 125o
3. In the figure, ABCD is a parallelogram. M C. 145o
is the midpoint of CD , Y is a point on AD D. 160o
and AX meets BY at X. If AX : XM = 1 : 2
and XY = 3, find the length of BX.
A B 6. In the square, x =
A . 9 X
B . 12 Y 3 A. 10
C . 15 B. 15
D . 21 C. 22.5
D M C
D. 30
9. If WXYZ is a rhombus, find e and f. (4 marks) 11. In the figure, ABCD is a rhombus. If the diagonal 12. In the Figure, ABCD is a parallelogram. EBDF
AC = 6 cm, CBE = 30, find is a straight line and EB = DF.
(a) BC,
(b) the area of the rhombus. (5 marks)

(a) Prove that ABE = CDF. (2 marks)

(b) Prove that EA // CF. (4 marks)

10. PQRS is a rhombus. If PR = 5 cm and QS = 10 cm.

Find the area of the rhombus. (5 marks)
P Q

R
Chapter 08 Properties of Quadrilaterals Quiz 08-3
F.3_____ Name:___________________________________________________( ) Marks: _____ / 28
1. In the figure, KLMN is a parallelogram and 2. In the figure, AB = BC. AD is the angle bisector of 3. In the figure, ABCD is a parallelogram. DFEB is a
QLK = PNM. QKMP is a straight line. Prove PDQ and PD = DQ. straight line and DF = EB. Show that AECF is a
that Prove that parallelogram. State ALL your reasons clearly.
(a) QLK and PNM are congruent, and (a) PB = QB, and (7 marks)
(b) LQNP is a parallelogram. (5 marks) (b) APCQ is a parallelogram. (5 marks)
4. In the figure, P, Q, R and S are the midpoints of AB, 5. In the figure, ABCD and AGFE are straight lines. If 7. In the figure, P is the midpoint of side BC of ABC.
BC, CD and DA respectively. Show that PQRS is a BC = 2 cm, CD = 3 cm, BG = 6 cm and CF = 10 cm, Q is the midpoint of AP. BQ is produced to meet
parallelogram. (5 marks) then DE = AC at R.
1
Prove that AR = 2 RC. (4 marks)
A. 12 cm
B. 14 cm [Hint: Draw PS parallel to QR to meet AC at S.]
C. 15 cm
D. 16 cm

6. In the figure, OABC and OFED are straight lines.

If AB : BC = 2 : 3 and OF : OD = 1 : 5, then
OA : AB =

A. 1:1
B. 1:2
C. 5:8
D. 5 : 13
Chapter 08 Properties of Quadrilaterals Quiz 08-S
F.3_____ Name:___________________________________________________( ) Marks: _____ /
1. In the figure, ABCD is a parallelogram. E and F are points on AB and CD 2. In the figure, ABCD is a square.
respectively such that AED = CFB. Prove that BEDF is a parallelogram.
(3 marks)

If AS = BP = CQ = DR, prove that PQRS is a square.

(8 marks)