parallelogram 平行四邊形 opposite side 對邊 opposite angle 對角 diagonal 對角線 bisect 平分 equilateral triangle 等邊三角形
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Lesson Worksheet 8.1A(II)
Objective: To understand and use the properties of parallelograms.
A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
平行四邊形是一個有兩對平行線的四邊形。
The followings are the properties of a parallelogram: 以下是平行四邊形的性質︰
(1) The opposite sides are equal. 對邊相等。
[Ref.: opp. sides of // gram] 〔簡記︰平行四邊形對邊〕
(2) The opposite angles are equal. 對角相等。
[Ref.: opp. s of // gram] 〔簡記︰平行四邊形對角〕
(3) The diagonals bisect each other. 對角線互相平分。
[Ref.: diags. of // gram] 〔簡記︰平行四邊形對角線〕
1. In the figure, ABCD is a parallelogram. Find the values of x and y.
∵ AD // BC (definition of // gram) DAC = BCA (alt. s, AD // BC)
ADC = ABC = (opp. s of // gram)
In △ACD,
ADC + DAC + ACD = 180 ( sum of △)
2. In each of the following, ABCD is a parallelogram. Find the unknown(s).
(a) AEC and BED are straight lines.
∵ DE = (diags. of // gram)
∵ CE = (diags. of // gram)
Name: ____________________ ( ) Class: Date:
117 + 35 + y = 180
y = 28
Demonstration
In the figure, ABCD is a parallelogram. Find the values of x and y.
Solution
AB // DC (definition of // gram)
CDB = ABD (alt. s, AB // DC) x = 70
In △BCD,
BDC + BCD + CBD = 180 ( sum of
△)
70 + y + 40 = 180
B C
A D
70 40
x y B C
A D
117 35
x
y
x = 35
117
7y – 2= 2y + 3 5y = 5
y = 1
B C
A 2y + 3 D
2x + 1 x + 5
7y – 2 E
2x + 1= x + 5 x = 4 BE
AE
parallelogram 平行四邊形 opposite side 對邊 opposite angle 對角 diagonal 對角線 bisect 平分 equilateral triangle 等邊三角形
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(b)
In △ACD,
∵ AD = CD (given)
∴ DCA = DAC = (base s, isos. △)
ADC + DAC + DCA = 180 ( sum of △)
∵ ABC = (opp. s of // gram)
3. In the figure, ABCD is a parallelogram and
△CDE is an equilateral triangle. BCE is a straight line. Find the values of x and y.
∵ AB = (opp. sides of // gram)
DCE = (property of equil. △)
BCD + = (adj. s on st. line)
∵ BAD = (opp. s of // gram)
x = 108
C B
A 36 D
x
36
ADC + 36 + 36 = 180
ADC = 108
y = 120
B C
A D
E
3x – 5 y 10
Demonstration
In the figure, ABCD is a parallelogram and △ CDE is an isosceles triangle. BCE is a straight line. Find the values of x and y.
Solution
BCD = BAD = 108 (opp. s of // gram) 108 + DCE = 180 (adj. s on st. line)
DCE = 72
In △CDE,
∵ DC = DE (given)
∴ DEC = DCE (base s, isos. △) x = 72
AB = DC (opp. sides of // gram) 2y + 3 = 12
y = 4.5
B C
A D
E 108
x
2y + 3 12
