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Published on: 16/09/2019
Quadrilaterals
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1.
In a parallelogram, show that the angle bisectors of two adjacent angles intersect at right angle.
2.
Two parallel lines I and m are intersected by a transversal ' t'.Show that the quadrilateral formed by bisectors of interior angles is a rectangle.
3.
If angles of a quadrilateral are in ratio 1 : 2 : 3 : 4. Find the measure of all the angles of a quadrilateral.
4.
Two opposite angles of a parallelogram are (3x - 2)° and (63 - 2x)° Find all the angles of a parallelogram.
5.
The angles of a quadrilateral are (4x°),(7x°),(15x°) and (10x°).Find the smallest and largest angles of the quadrilateral.
6.
In \(\Delta \) ABC, D, E and F are midpoints of sides AB, BC and CA. If AB = 6 cm,BC = 7.2 cm and AC = 7.8 crn find the perimeter of \(\Delta \)DEF.
7.
In the figure, ABCD is a parallelogram in which AB is produced to E so that AB = BE
(a) Prove that ED bisects BC
(b) If AD = 10 cm, find OB.
8.
If an angle of a parallelogram in two-third of its adjacent angle then find the measure of all the angles,
9.
In a parallelogram PQRS, if \(\angle \)QRS=2x, \(\angle \)PQS=4x, and \(\angle \)PSQ=4x, find the angles of the parallelogram.
10.
The angles A, B, C and D of a quadrilateral have measures in the ratio 2 : 4 : 5 : 7. Find the measures of these angles. What type of quadrilateral is it? Give reasons.
1.
ㄥADC+ㄥBCD=180°
⇒ \(\frac{1}{2}\)ㄥADC+\(\frac{1}{2}\)ㄥBCD=90°
or ㄥ1+ㄥ2=90°
In ΔODC,
ㄥ1+ㄥ2+ㄥDOC=180°
ㄥDOC=90°
2.
∠APR=ㄥDRP
or ㄥ1=ㄥ2
But these are alternate interior angles
SP II RQ, SR II PQ
PQRS is a parallelogram
∠APR+ㄥBPR=180°,(linear pair)
⇒ \(\frac{1}{2}\) ∠APR+\(\frac{1}{2}\)ㄥBPR=\(\frac{1}{2}\)x180°
⇒ ∠1+ㄥ3=90°
⇒ ∠SPQ=90°
PQRS is a rectangle
3.
Let the measure of the angles be x, 2.x, 3x and 4x then,
x + 2x + 3x + 4x = 360°
⇒ x = 36°
ஃ Angles of quadrilateral are 36°, 72°, 108°,144°
4.
Since opposite angles of a parallelogram are equal
3x - 2 = 63 - 2x
⇒ x=13°
Angles of a parallelogram :
(39 - 2)°, (180- 37)°, (63 - 26)°, (180- 37)°
i.e., 37°,143°,37°,143°.
5.
Sum of the angles of a quadrilateral is 360°.
ஃ 4x° + 7x° + 15x° + 10x° = 360°
[Angle sum property of quadrilateral]
⇒ 36x°= 360°
⇒ x=10°
ஃ Smallest angle = 4x° = 40°
Largest angle = 15x°= 150°
6.
10.5 cm
7.
5 cm
8.
72°, 108°, 72°, 108°
9.
\(36°\),\(144°\),\(36°\), \(144°\)
10.
40°, 80°, 100°, 140°; Trapezium
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