**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:**

- Number of sides of a quadrilateral = 4

Number of angles of a quadrilateral = 4

Number of diagonals of a quadrilateral = 2 - The sum of the measures of four angles of a quadrilateral is 360°.
- 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° - 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°.