Introduction To Geometry
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F.1 Mathematics Supplementary Notes
Chapter 8 & 10
Chapter 8 Introduction to Geometry Chapter 10 Congruence and Similarity Important Terms Geometry line segment end point point of intersection acute angle right angle obtuse angle congruent to
幾何學 線段 端點 交點 銳角 直角 鈍角 全等於
4/2002
Name:___________( )Class: F.1 ( )
isosceles triangle equilateral triangle polygon regular polygon parallel lines perpendicular lines diagonal proportional
等腰三角形 等邊三角形 多邊形 正多邊形 平行線 垂直線 對角線 成比例
centre circumference diameter radius solid cross-section vertices similar to
圓心 圓周 直徑 半徑 立體 橫切面 頂點 相似於
Exercise 1 1. Refer to the figure A D
(a) (b) (c) (d)
C B Produce(延長) BA and CD to meet at P. Produce BC to Q so that the length of CQ is less than that of BC. Join AQ to meet the line segment CD at M. Join the two diagonals AC and BD，and mark the point of intersection by F.
2. Refer to the figure Angle (a) (b) (c) (d) (e) (f)
∠ ABF ∠ EBA ∠ FBC b ∠ ABC ∠ EBD
Kind acute angle ___________ ___________ ___________ ___________ ___________
E b
D 250
F
400
C
B A
A
D E B
700
b C
d
P. 1
F.1 Mathematics Supplementary Notes
3.
Chapter 8 & 10
4/2002
In the figure, AB＝BC＝CA＝CD. (a) The isosceles triangles are _________________________ and an equilateral triangle is _________________________ (b) ∠ ACB＝__________________ (c ) b + d = __________________
4.
Find the unknown in each of the following triangles. (a) (b) 29 0 27 0
3x
x–5 0
x
a
(c)
(d)
48° 41 0
b
17° x
*5. In the figure DCA=DCB , DBA=DBC , BAC=64°, find x.
A 64° D x B
C
P. 2
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 3
Exercise 2 1.
In each of the following pair of triangles, decide whether ∆ ABC is congruent to the other triangle or not. For each pair of congruent triangles, give the abbreviated reference.
(a)
(b)
AB = FD BC = DE AC = FE ∴ ∆ ABC ≅ ∆ FDE (SSS) (c)
(d)
(e)
(f)
F.1 Mathematics Supplementary Notes
(g)
2.
Chapter 8 & 10
4/2002
(h)
In each of the following, find out which two of the given triangles are congruent and give the abbreviated reason.
P. 4
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 5
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
Conditions for Similar Triangles ∠BAC=∠QPR ∠ABC=∠PQR ∠BCA=∠QRP (equiangular) ( or AAA)
(3 sides prop.)
∠ACB=∠PRQ (2 sides prop. inc. ∠ eq.)
Exercise 3 1. In each of the following, state whether the given pair of triangles are similar or not. For each pair of similar triangles, give the abbreviated reference.
Reasoning Steps ∆ ABC ∼ ∆ RPQ (3 sides prop.)
CB 5 1 = = QP 15 3
BA 4 1 = = PR 12 3 CA 3 1 = = QR 9 3 CB BA CA 1 = = = QP PR QR 3
∴ ∆ ABC ∼ ∆ RPQ (3 sides prop.)
P. 6
F.1 Mathematics Supplementary Notes
1.
Chapter 8 & 10
3.
C
A
C
Q
R C
5.
B X
X
Z
C
4.
P
B
Y
Y
Z B
A
2.
X Z
A
4/2002
Q A
6. A
B C
Z
XP
B Y C A Z
Y
B
R
P. 7
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
7.
8.
9.
10.
11.
4/2002
12.
P. 8
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
13.
14.
15.
16.
17. Given that ∆ ABC ∼ ∆ XYZ, ∠ BAC = 1230, ∠ ABC = 250, AB = 6cm, XY = 2cm, YZ = 3.2cm, find
(a) ∠ XYZ
(b) ∠ YZX
(c) BC.
P. 9
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 10
18. Given that ∆ ABC ∼ ∆ DEF, AB = 3cm, AC = 5cm, BC = 7cm, EF = 8.4cm, ∠ BAC = 1200, find (a) ∠ FDE
(b) DE
(c) DF.
Level II 19. In each of the following, find out which two of the given triangles are similar and prove it.
20.
In each of the following, find out a pair of similar triangles and prove it, hence find the value(s) of the marked unknowns. (a)
(b)
F.1 Mathematics Supplementary Notes
(c
Chapter 8 & 10
4/2002
)
(d)
(e)
P. 11
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 12
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 13
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 14
Chapter 8 & 10
Chapter 8 Introduction to Geometry Chapter 10 Congruence and Similarity Important Terms Geometry line segment end point point of intersection acute angle right angle obtuse angle congruent to
幾何學 線段 端點 交點 銳角 直角 鈍角 全等於
4/2002
Name:___________( )Class: F.1 ( )
isosceles triangle equilateral triangle polygon regular polygon parallel lines perpendicular lines diagonal proportional
等腰三角形 等邊三角形 多邊形 正多邊形 平行線 垂直線 對角線 成比例
centre circumference diameter radius solid cross-section vertices similar to
圓心 圓周 直徑 半徑 立體 橫切面 頂點 相似於
Exercise 1 1. Refer to the figure A D
(a) (b) (c) (d)
C B Produce(延長) BA and CD to meet at P. Produce BC to Q so that the length of CQ is less than that of BC. Join AQ to meet the line segment CD at M. Join the two diagonals AC and BD，and mark the point of intersection by F.
2. Refer to the figure Angle (a) (b) (c) (d) (e) (f)
∠ ABF ∠ EBA ∠ FBC b ∠ ABC ∠ EBD
Kind acute angle ___________ ___________ ___________ ___________ ___________
E b
D 250
F
400
C
B A
A
D E B
700
b C
d
P. 1
F.1 Mathematics Supplementary Notes
3.
Chapter 8 & 10
4/2002
In the figure, AB＝BC＝CA＝CD. (a) The isosceles triangles are _________________________ and an equilateral triangle is _________________________ (b) ∠ ACB＝__________________ (c ) b + d = __________________
4.
Find the unknown in each of the following triangles. (a) (b) 29 0 27 0
3x
x–5 0
x
a
(c)
(d)
48° 41 0
b
17° x
*5. In the figure DCA=DCB , DBA=DBC , BAC=64°, find x.
A 64° D x B
C
P. 2
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 3
Exercise 2 1.
In each of the following pair of triangles, decide whether ∆ ABC is congruent to the other triangle or not. For each pair of congruent triangles, give the abbreviated reference.
(a)
(b)
AB = FD BC = DE AC = FE ∴ ∆ ABC ≅ ∆ FDE (SSS) (c)
(d)
(e)
(f)
F.1 Mathematics Supplementary Notes
(g)
2.
Chapter 8 & 10
4/2002
(h)
In each of the following, find out which two of the given triangles are congruent and give the abbreviated reason.
P. 4
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 5
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
Conditions for Similar Triangles ∠BAC=∠QPR ∠ABC=∠PQR ∠BCA=∠QRP (equiangular) ( or AAA)
(3 sides prop.)
∠ACB=∠PRQ (2 sides prop. inc. ∠ eq.)
Exercise 3 1. In each of the following, state whether the given pair of triangles are similar or not. For each pair of similar triangles, give the abbreviated reference.
Reasoning Steps ∆ ABC ∼ ∆ RPQ (3 sides prop.)
CB 5 1 = = QP 15 3
BA 4 1 = = PR 12 3 CA 3 1 = = QR 9 3 CB BA CA 1 = = = QP PR QR 3
∴ ∆ ABC ∼ ∆ RPQ (3 sides prop.)
P. 6
F.1 Mathematics Supplementary Notes
1.
Chapter 8 & 10
3.
C
A
C
Q
R C
5.
B X
X
Z
C
4.
P
B
Y
Y
Z B
A
2.
X Z
A
4/2002
Q A
6. A
B C
Z
XP
B Y C A Z
Y
B
R
P. 7
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
7.
8.
9.
10.
11.
4/2002
12.
P. 8
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
13.
14.
15.
16.
17. Given that ∆ ABC ∼ ∆ XYZ, ∠ BAC = 1230, ∠ ABC = 250, AB = 6cm, XY = 2cm, YZ = 3.2cm, find
(a) ∠ XYZ
(b) ∠ YZX
(c) BC.
P. 9
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 10
18. Given that ∆ ABC ∼ ∆ DEF, AB = 3cm, AC = 5cm, BC = 7cm, EF = 8.4cm, ∠ BAC = 1200, find (a) ∠ FDE
(b) DE
(c) DF.
Level II 19. In each of the following, find out which two of the given triangles are similar and prove it.
20.
In each of the following, find out a pair of similar triangles and prove it, hence find the value(s) of the marked unknowns. (a)
(b)
F.1 Mathematics Supplementary Notes
(c
Chapter 8 & 10
4/2002
)
(d)
(e)
P. 11
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 12
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 13
F.1 Mathematics Supplementary Notes
Chapter 8 & 10
4/2002
P. 14
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