问题1.给定平行四边形ABCD。完成每个语句以及所使用的定义或属性。
(i)AD =……
(ii)∠DCB=……
(iii)OC =……
(iv)m∠DAB+ m∠CDA=……
解决方案：
(i) AD = BC {Opposite sides of a parallelogram are equal}
(ii) ∠DCB = ∠DAB {Opposite angles of a parallelogram are equal}
(iii) OC = OA {Diagonals of a parallelogram are equal}
(iv) m ∠DAB + m ∠CDA = 180°
问题2.考虑以下平行四边形。查找未知的x，y，z的值
解决方案：
(一世)
y = 100° {opposite angles of a parallelogram}
x + 100° = 180° {Adjacent angles of a parallelogram}
⇒ x = 180° – 100°
⇒ x = 80°
x = z = 80° {opposite angles of a parallelogram}
Therefore,
x = 80°, y = 100° and z = 80°
(ii)
50° + x = 180°
⇒ x = 180° – 50° = 130° {Adjacent angles of a parallelogram}
⇒ x = y = 130° {opposite angles of a parallelogram}
⇒ x = z = 130° {corresponding angle}
(iii)
x = 90° {vertical opposite angles}
x + y + 30° = 180° {angle sum property of a triangle}
⇒ 90° + y + 30° = 180°
⇒ y = 180° – 120° = 60°
also, y = z = 60° {alternate angles}
(iv)
z = 80° {corresponding angle}
z = y = 80° {alternate angles}
x + y = 180° {adjacent angles}
⇒ x + 80° = 180°
⇒ x = 180° – 80° = 100°
(v)
y = 112° {opposite angles of a parallelogram}
x = 180° – (y + 40°) {angle sum property of a triangle}
x = 28°
z = 28° {alternate angles}
问题3.如果四边形ABCD是平行四边形，如果
(i)∠D+∠B= 180°?
(ii)AB = DC = 8厘米，AD = 4厘米，BC = 4.4厘米?
(iii)∠A= 70°，andC = 65°?
解决方案：
(i) Yes,
The quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°,
it should also fulfilled some conditions which are:
(a) The sum of the adjacent angles should be 180°.
(b) Opposite angles must be equal.
(ii) No, opposite sides should be of the same length.
Here, AD ≠ BC
(iii) No, opposite angles should be of same measures.
Here, ∠A ≠ ∠C
问题4.画出四边形的粗略图，该图不是平行四边形，而是正好有两个相等的对角。
解决方案：
ABCD is a figure of quadrilateral which is not a parallelogram but has exactly two opposite angles
that is ∠B = ∠D of equal measure. It is not a parallelogram because ∠A ≠ ∠C.
问题5：平行四边形的两个相邻角度的比例为3：2。找到平行四边形的每个角度的度量。
解决方案：
Let the measures of two adjacent angles ∠A and ∠B be 3x and 2x respectively in parallelogram ABCD.
∠A + ∠B = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 36°
As we know opposite sides of a parallelogram are equal.
∠A = ∠C = 3x = 3 × 36° = 108°
∠B = ∠D = 2x = 2 × 36° = 72°
问题6.平行四边形的两个相邻角度具有相等的度量。找到平行四边形的每个角度的度量。
解决方案：
Let ABCD be a parallelogram.
The sum of adjacent angles of a parallelogram = 180°
∠A + ∠B = 180°
⇒ 2∠A = 180°
⇒ ∠A = 90°
also,
90° + ∠B = 180°
⇒ ∠B = 180° – 90° = 90°
∠A = ∠C = 90°
∠B = ∠D = 90°
问题7.相邻的图形HOPE是平行四边形。找到角度量度x，y和z。陈述您用于查找它们的属性。
解决方案：
y = 40° {alternate interior angle}
∠P = 70° {alternate interior angle}
∠P = ∠H = 70° {opposite angles of a parallelogram}
z = ∠H – 40°= 70° – 40° = 30°
∠H + x = 180°
⇒ 70° + x = 180°
⇒ x = 180° – 70° = 110°
问题8.下图GUNS和RUNS是平行四边形。找出x和y。 (长度以厘米为单位)
解决方案：
(一世)
SG = NU and SN = GU {opposite sides of a parallelogram are equal}
3x = 18
x = 18/3
⇒ x =6
3y – 1 = 26 and,
⇒ 3y = 26 + 1
⇒ y = 27/3=9
x = 6 and y = 9
(ii)
20 = y + 7 and 16 = x + y {diagonals of a parallelogram bisect each other}
y + 7 = 20
⇒ y = 20 – 7 = 13 and,
x + y = 16
⇒ x + 13 = 16
⇒ x = 16 – 13 = 3
x = 3 and y = 13
问题9.在上图中， RISK和CLUE均为平行四边形。求x的值。
解决方案：
∠K + ∠R = 180° {adjacent angles of a parallelogram are supplementary}
⇒ 120° + ∠R = 180°
⇒ ∠R = 180° – 120° = 60°
also, ∠R = ∠SIL {corresponding angles}
⇒ ∠SIL = 60°
also,
∠ECR = ∠L = 70° {corresponding angles}
x + 60° + 70° = 180° {angle sum of a triangle}
⇒ x + 130° = 180°
⇒ x = 180° – 130° = 50°
问题10.解释这个数字是不是梯形。它的哪一侧是平行的? (数字)
解决方案：
When a transversal line intersects two lines in such a way that the sum of the adjacent angles on the same side of transversal is 180° then, the lines are parallel to each other.
Here we have, ∠M + ∠L = 100° + 80° = 180°
Hence, MN || LK
As the quadrilateral KLMN has one pair of parallel line therefore it is a trapezium.
MN and LK are parallel lines.
问题11：如果AB ||，在图中找到m∠C DC?
解决方案：
m∠C + m∠B = 180° {angles on the same side of transversal}
⇒ m∠C + 120° = 180°
⇒ m∠C = 180° − 120° = 60°
问题12：如果SP ||，求FindP和∠S的量度RQ?在图中。 (如果找到m∠R，找到m∠P的方法不止一种吗?)
解决方案：
∠P + ∠Q = 180° {angles on the same side of transversal}
⇒ ∠P + 130° = 180°
⇒ ∠P = 180° – 130° = 50°
also,
∠R + ∠S = 180° {angles on the same side of transversal}
⇒ 90° + ∠S = 180°
⇒ ∠S = 180° – 90° = 90°
Hence, ∠P = 50° and ∠S = 90°
Yes, there are more than one method to find m∠P.
PQRS is a quadrilateral. Sum of measures of all angles of a quadrilateral is 360°.
Thus, as we know the measurement of ∠Q, ∠R and ∠S.
∠Q = 130°, ∠R = 90° and ∠S = 90°
∠P + 130° + 90° + 90° = 360°
⇒ ∠P + 310° = 360°
⇒ ∠P = 360° – 310° = 50°