问题1.四边形的三个角度分别等于110°，50°和40°。找到它的第四个角度。
解决方案：
Given,
The three angles are 110°, 50° and 40°
Let the fourth angle be ‘x’
Sum of all angles of a quadrilateral = 360°
110° + 50° + 40° + x = 360°
200° + x = 360°
x = 360° − 200°
x = 160°
Hence, the required fourth angle is 160°.
问题2.在四边形ABCD中，角度A，B，C和D的比例为1：2：4：5。找到四边形的每个角度的度量。
解决方案：
Let the angles of quadrilateral be
A = x, B = 2x, C = 4x, D = 5x
Then,
A + B + C + D = 360° {Sum of interior angle of quadrilateral 360°}
x + 2x + 4x + 5x = 360°
12x = 360°
x = 360°
12
x = 30°
Therefore, substituting value of x,
A = x = 30°
B = 2x = 2 × 30° = 60°
C = 4x = 4 × 30° = 120°
D = 5x = 5 × 30° = 150°
问题3.在四边形ABCD中，CO和Do分别是∠C和∠D的等分线。证明∠COD= 1/2 ( ∠A和∠B)
解决方案：
In ΔDOC,
∠CDO + ∠COD + ∠DCO = 1800 [Angle sum property of a triangle]
or
1 ∠CDA + ∠COD + 1 ∠DCB = 1800
2 2
∠COD = 1800 – 1 (∠CDA + ∠DCB) —->(equation 1)
2
Also
We know, sum of all angles of a quadrilateral = 360°
∠CDA + ∠DCB = 3600 – (∠DAB + ∠CBA) —->(equation 2)
Substituting equation 1 and equation 2
∠COD = 1800 – 1 {3600 – (∠DAB + ∠CBA)}
2
We can also write,
∠DAB = ∠A and ∠CBA = ∠B
∠COD = 1800 − 1800 + 1/2(∠A + ∠B))
∠COD = 1/2(∠A + ∠B)
Hence, Proved.
问题4：四边形的角度比例为3：5：9：13。找到所有四边形的角度。
解决方案：
Let the angles of quadrilateral be
A = 3x, B = 5x, C = 9x, D = 13x
Then,
A + B + C + D = 360° {{Sum of interior angle of quadrilateral 360°}
3x + 5x + 9x + 13x = 360°
30x = 360°
x = 360°
30
x = 12°
Therefore, substituting value of x,
A = 3x = 3×12 = 36°
B = 5x = 5×12° = 60°
C = 9x = 9×12° = 108°
D = 13x = 13×12° = 156°