GSEB Solutions for Class 8 Mathematics – Quadrilateral

GSEB Solutions for Class 8 Mathematics – Quadrilateral (English Medium)

GSEB SolutionsMathsScience
Exercise

Solution 1:

DEGF can be named in eight different ways as follows:
DEGF, EGFD, GFDE, FDEG, DFGE, FGED, GEDF, EDFG.

Solution 2:

Solution 3:
In STUV, m∠S = 90°, m∠V = 80° and
let m∠T = m∠U = x
Now, in STUV,
m∠S + m∠T + m∠U + m∠V = 360°
∴ 90° + x + x + 80° = 360°
∴ 2x + 170° = 360°
∴ 2x = 360° – 170°
∴ 2x = 190°
∴ x = 190°/2
∴ x = 95°
∴ m∠T = m∠U = 95°
Thus, in STUV, m∠T = 95° and m∠U = 95°.

Solution 4:
In PQRS, let m∠P = x.
Then, we have
m∠Q = x – 10°
m∠R = x – 20°
m∠S = x – 30°
In PQRS,
m∠P + m∠Q + m∠R + m∠S = 360°
∴ x + (x – 10°) + (x – 20°) + (x – 30°) = 360°
∴ x + x – 10° + x – 20° + x – 30° = 360°
∴ 4x – 60° = 360°
∴ 4x = 360° + 60°
∴ 4x = 420°
∴ x = 420°/4
∴ x = 105°
Then, m∠P = x = 105°
m∠Q = x – 10° = 105° – 10° = 95°
m∠R = x – 20° = 105° – 20° = 85°
and m∠S = x – 30° = 105° – 30° = 75°
Thus, in PQRS, the measures of four angles are:
m∠P = 105°, m∠Q = 95°, m∠R = 85° and m∠S = 75°.

Solution 5:
The sum of measures of all the angles of a quadrilateral is 360°.
The measure of one of the angles is 120°.
∴ The sum of the measures of the remaining three angles
= 360° – 120° = 240°
Now, the measures of the remaining three angles are equal.
∴ The measure of each of the remaining angles
= 240°/3  = 80°
Thus, the measure of each of the remaining three angles is 80°.

Solution 6:

1. Number of sides of a quadrilateral = 4
   Number of angles of a quadrilateral = 4
   Number of diagonals of a quadrilateral = 2
2. The sum of the measures of four angles of a quadrilateral is 360°.
3. In a square, all the four angles are right angles, i.e. each angle measures 90°.
   ∴The sum of the measures of the three angles of a square
   = 90° + 90° + 90°
   = 270°
4.  A square and a rhombus have equal measures of all the four sides.

Practice 1

Solution 1:

Solution 2:
□PQRS, □QRSP, □RSPQ, □SPQR, □PSRQ, □SRQP, □RQPS, □QPSR

□XYZW, □YZWX, □ZWXY, □WXYZ, □XWZY, □WZYX, □ZYXW, □YXWZ

□LMNO, □MNOL, □NOLM, □OLMN, □LONM, □ONML, □NMLO, □MLON

Solution 3:
A quadrilateral with vertices S, T, U and V can be given a name using two methods; clockwise and anticlockwise.

Clockwise direction: □SVUT, □VUTS, □UTSV, □TSVU
Anti-clockwise direction: □STUUV, □TUVS, □UVST, □VSTU

Practice 2

Solution 1:

Practice 3

Solution 1:
The sum of the measures of the three angles of a quadrilateral
= 75° + 65° + 120°
= 260°
The sum of the measures of all four angles of a quadrilateral is 360°.
∴ Measure of the fourth angle = 360° – 260° = 100°
Thus, the measure of the fourth angle of the quadrilateral is 100°.

Solution 2:
The sum of measures of two angles of a quadrilateral = 80° + 100° = 180°.
The sum of the measures of all four angles of a quadrilateral is 360°.
∴ The sum of the measures of other two angles = 360° – 180° = 180°
Since the measures of these two angles are equal, measure of each angle = 180° ÷ 2 = 90°
Thus, the measure of each of the angles having equal measures is 90°.

Solution 3:
In □MNOP, let m∠M = x.
∴ m∠N = x + 10°
m∠O = x + 20°
m∠P = x + 30°.
For □MNOP,
m∠M + m∠N + m∠O + m∠P = 360°
∴ x + x + 10° + x + 20° + x + 30° = 360°
∴ 4x + 60° = 360°
∴ 4x = 360° – 60°
∴ 4x = 300°
∴ x = 300°/4
∴ x = 75°
Now, m∠M = x = 75°,
m∠n = x + 10° = 75° + 10° = 85°,
m∠O = x + 20° = 75° + 20° = 95°
m∠P = x + 30° = 75° + 30° = 105°
Thus, the measures of the angles of □MNOP are
m∠M = 75°, m∠N = 85°, m∠O = 95° and m∠P = 105°.

Solution 4:
In □DEFG, m∠D = 120°, m∠F = 140°
Let m∠E = m∠G = x (say).
In □DEFG,
m∠D + m∠E + m∠F + m∠G = 360°
∴ 120° + x + 140° + x = 360°
∴ 2x + 260° = 360°
∴ 2x = 360° – 260°
∴ 2x = 100°
∴ x = 100°/2
∴ x = 50°
∴ m∠E = m∠G = 50°.
Thus, in □DEFG, m∠E = 50° and m∠G = 50°.

Solution 5:
Sum of measures of all the four angles of a quadrilateral is 360°.
Here, the measures of all four angles are equal.
∴ Measure of each angle of a quadrilateral
= 360°/4
= 90°
Thus, the measure of each angle of the quadrilateral is 90°.

Solution 6:
Sum of the measures of all the angles of a quadrilateral is 360°.
One angle of a quadrilateral is a right angle.
∴ The measure of this angle is 90°.
The measure of one other angle of a quadrilateral is 110°.
So, the sum of measures of the two given angles of the quadrilateral = 90° + 110° = 200°.
∴ The sum of measures of the remaining two angles of the quadrilateral = 360° – 200° = 160°.
The measures of these two remaining angles are equal.
∴ Measure of each of the remaining two angles
= 160°/2
=  80°
Thus, the measure of each of the angles with equal measures is 80°.