NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 4 — Figure it Out (Page 102)#
- In a parallelogram, opposite angles are equal and adjacent angles add up to $180°$. Its diagonals bisect each other.
- A rhombus is a parallelogram with all four sides equal. Its diagonals bisect each other at $90°$.
1. Find the remaining angles in the following quadrilaterals. #
(i) Parallelogram PEAR with ∠P = 40°
PE ∥ RA and PR ∥ EA, so PEAR is a parallelogram. Adjacent angles add up to $180°$:
$\angle E = 180° - \angle P$
$\angle E = 180° - 40°$
$\angle E = 140°$
Opposite angles are equal:
$\angle A = \angle P = 40°$
$\angle R = \angle E = 140°$
∴ $\angle E = 140°$, $\angle A = 40°$ and $\angle R = 140°$.
(ii) Parallelogram PQRS with ∠P = 110°
PQ ∥ SR and PS ∥ QR, so PQRS is a parallelogram. Adjacent angles add up to $180°$:
$\angle Q = 180° - \angle P$
$\angle Q = 180° - 110°$
$\angle Q = 70°$
Opposite angles are equal:
$\angle R = \angle P = 110°$
$\angle S = \angle Q = 70°$
∴ $\angle Q = 70°$, $\angle R = 110°$ and $\angle S = 70°$.
(iii) Rhombus UVWX with ∠XVU = 30°
All four sides are marked equal, so UVWX is a rhombus.
In $\triangle XUV$, $UX = UV$, so the angles opposite them are equal:
$\angle UXV = \angle UVX = 30°$
$\angle U = 180° - (30° + 30°)$
$\angle U = 120°$
In $\triangle XWV$, $WX = WV$ in the same way, and the triangles $XUV$ and $XWV$ are congruent (SSS), so:
$\angle WXV = \angle WVX = 30°$
$\angle W = \angle U = 120°$
The diagonal XV splits the angles at X and V into two $30°$ parts:
$\angle X = 30° + 30° = 60°$
$\angle V = 30° + 30° = 60°$
∴ $\angle U = \angle W = 120°$ and $\angle X = \angle V = 60°$, with $\angle UXV = \angle WXV = \angle WVX = 30°$.
(iv) Rhombus OIEA with ∠OEA = 20°
All four sides are marked equal, so OIEA is a rhombus.
In $\triangle OAE$, $AO = AE$, so:
$\angle AOE = \angle AEO = 20°$
$\angle A = 180° - (20° + 20°)$
$\angle A = 140°$
In $\triangle OIE$, $IO = IE$, and the triangle is congruent to $\triangle OAE$ (SSS), so:
$\angle IOE = \angle IEO = 20°$
$\angle I = \angle A = 140°$
The diagonal OE splits the angles at O and E:
$\angle O = 20° + 20° = 40°$
$\angle E = 20° + 20° = 40°$
∴ $\angle A = \angle I = 140°$ and $\angle O = \angle E = 40°$, with $\angle AOE = \angle IOE = \angle IEO = 20°$.
2. Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°. #
Construction
The diagonals of a parallelogram bisect each other, so the halves are $3.5$ cm and $2.5$ cm.
- Draw a line segment $AB = 7$ cm and mark its midpoint O ($OA = OB = 3.5$ cm).
- At O, draw a line making an angle of $140°$ with OB.
- With centre O and radius $2.5$ cm, cut this line at C and D on either side of O, so $CD = 5$ cm.
- Join A to C, C to B, B to D and D to A.
ADBC is the required parallelogram, because its diagonals AB and CD bisect each other.
3. Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm. #
Construction
The diagonals of a rhombus bisect each other at right angles, so the halves are $2$ cm and $2.5$ cm, and they meet at $90°$.
- Draw a line segment $AB = 5$ cm and mark its midpoint O.
- Draw the line through O perpendicular to AB (the perpendicular bisector of AB, using a compass).
- With centre O and radius $2$ cm, cut this line at C and D on either side of O, so $CD = 4$ cm.
- Join A to C, C to B, B to D and D to A.
ADBC is the required rhombus. Every side is the hypotenuse of a right triangle with legs $2.5$ cm and $2$ cm, so all four sides are equal.