Understanding Quadrilaterals

Based on standard properties:
(a) Simple curve: A curve that does not cross itself.
(b) Simple closed curve: A simple curve whose starting and endpoints meet.
(c) Polygon: A simple closed curve made up of only line segments.
(d) Convex polygon: A polygon where all interior angles are less than 180° and no diagonal goes outside.
(e) Concave polygon: A polygon with at least one interior angle greater than 180° and at least one diagonal going outside.

The formula for the number of diagonals in an n-sided polygon is: n(n - 3) / 2

(a) A convex quadrilateral:
Here, n = 4.
Number of diagonals = 4(4 - 3) / 2 = 4(1) / 2 = 2.

(b) A regular hexagon:
Here, n = 6.
Number of diagonals = 6(6 - 3) / 2 = 6(3) / 2 = 18 / 2 = 9.

(c) A triangle:
Here, n = 3.
Number of diagonals = 3(3 - 3) / 2 = 3(0) / 2 = 0.

The sum of the measures of the angles of a convex quadrilateral is 360°.
Yes, this property holds true even if the quadrilateral is not convex (i.e., a concave quadrilateral). Any quadrilateral can be divided into two triangles by drawing a diagonal from one vertex to the opposite non-adjacent vertex. Since the sum of angles in one triangle is 180°, for two triangles it will always be 180° + 180° = 360°.

The formula for the sum of interior angles of a convex polygon with n sides is: (n - 2) × 180°.

(a) For n = 7 (Heptagon):
Sum of angles = (7 - 2) × 180° = 5 × 180° = 900°.

(b) For n = 8 (Octagon):
Sum of angles = (8 - 2) × 180° = 6 × 180° = 1080°.

(c) For n = 10 (Decagon):
Sum of angles = (10 - 2) × 180° = 8 × 180° = 1440°.

(d) For n sides:
Sum of angles = (n - 2) × 180°.

A polygon having all sides of equal length and all interior angles of equal measure is known as a regular polygon. That is, it is both equilateral and equiangular.

(i) 3 sides: Equilateral Triangle.
(ii) 4 sides: Square.
(iii) 6 sides: Regular Hexagon.

(a) Sum of angles of a quadrilateral = 360°.
50° + 130° + 120° + x = 360°
300° + x = 360° ⇒ x = 60°.

(b) The interior angle adjacent to the exterior 90° angle is 180° - 90° = 90°.
Sum of angles of a quadrilateral = 360°.
x + 70° + 60° + 90° = 360°
x + 220° = 360° ⇒ x = 140°.

(c) The interior base angles are supplementary to the exterior angles.
First interior base angle = 180° - 70° = 110°.
Second interior base angle = 180° - 60° = 120°.
Sum of angles of a pentagon (5 sides) = (5 - 2) × 180° = 540°.
30° + x + x + 110° + 120° = 540°
260° + 2x = 540° ⇒ 2x = 280° ⇒ x = 140°.

(d) The given figure is a regular pentagon (all sides are marked equal).
Sum of angles of a pentagon = 540°.
Since it's regular, all 5 interior angles are equal to x.
5x = 540° ⇒ x = 540° / 5 = 108°.

(a) The sum of the measures of the exterior angles of any polygon is always 360°.
Therefore, x + y + z = 360°.
(Verification: Interior angle 1 = 90° ⇒ x = 90°. Interior angle 2 = 30° ⇒ z = 150°. Third interior angle = 180° - (90°+30°) = 60° ⇒ y = 120°. Sum = 90°+150°+120° = 360°.)

(b) The sum of the measures of the exterior angles of any convex polygon is always 360°.
Therefore, x + y + z + w = 360°.
(Verification: 4th interior angle = 360° - (60°+80°+120°) = 100°. Exterior angles are: 180°-120°=60°, 180°-80°=100°, 180°-60°=120°, 180°-100°=80°. Sum = 60°+100°+120°+80° = 360°.)

We know that the sum of the measures of the exterior angles of any polygon is always 360°.

(a) 125° + 125° + x = 360°
250° + x = 360°
x = 360° - 250° = 110°.

(b) The exterior angles are x, 90° (indicated by the square symbol), 60°, 90° (vertically opposite or supplementary to the interior 90°), and 70°.
x + 90° + 60° + 90° + 70° = 360°
x + 310° = 360°
x = 360° - 310° = 50°.

For a regular polygon, measure of each exterior angle = 360° / n, where n is the number of sides.

(i) 9 sides:
Each exterior angle = 360° / 9 = 40°.

(ii) 15 sides:
Each exterior angle = 360° / 15 = 24°.

Measure of each exterior angle = 24°.
Sum of exterior angles = 360°.
Number of sides (n) = 360° / (Each exterior angle)
n = 360° / 24° = 15 sides.

Interior angle = 165°.
Exterior angle = 180° - Interior angle
= 180° - 165° = 15°.
Number of sides (n) = 360° / Exterior angle
n = 360° / 15° = 24 sides.

(a) If exterior angle = 22°,
Number of sides (n) = 360° / 22° = 180 / 11 = 16.36...
Since the number of sides is not a whole number, no, it is not possible.

(b) If interior angle = 22°,
Exterior angle = 180° - 22° = 158°.
Number of sides (n) = 360° / 158° = 180 / 79.
Since this is also not a whole number, no, it cannot be an interior angle of a regular polygon either.

(a) A regular polygon with the minimum number of sides is an equilateral triangle (n = 3).
Its interior angle = (3 - 2) × 180° / 3 = 180° / 3 = 60°.
So, the minimum interior angle possible for a regular polygon is 60°.

(b) Since the minimum interior angle is 60°, the corresponding maximum exterior angle will be 180° - 60° = 120°.

(i) AD = BC. Property: In a parallelogram, opposite sides are equal in length.

(ii) ∠DCB = ∠DAB. Property: In a parallelogram, opposite angles are equal in measure.

(iii) OC = OA. Property: In a parallelogram, the diagonals bisect each other.

(iv) m∠DAB + m∠CDA = 180°. Property: In a parallelogram, adjacent angles are supplementary.

(Solving generally based on parallelogram properties)
- If one angle is 100°: The opposite angle is also 100°. The adjacent angles are 180° - 100° = 80°.
- If one angle is 50°: The opposite angle is 50°. The adjacent angles are 180° - 50° = 130°.
- If an exterior angle is involved: Use linear pairs along with parallelogram properties.
- If diagonals are given as expressions (like 20 = y + 7, 16 = x + y): Since diagonals bisect each other, equate the segments. From the first, y = 13. Substitute in the second: 16 = x + 13 ⇒ x = 3.

(i) Can be, but need not be. Opposite angles must be equal (∠D = ∠B), so each must be 90°. This makes it a rectangle (which is a parallelogram), but a quadrilateral with opposite angles summing to 180° could also be a cyclic quadrilateral without being a parallelogram.

(ii) No. In a parallelogram, opposite sides are equal. Here AD (4 cm) ≠ BC (4.4 cm).

(iii) No. In a parallelogram, opposite angles are equal. Here ∠A (70°) ≠ ∠C (65°).

A Kite provides a good example. In a kite, the angles formed by unequal sides are equal. Let ABCD be a kite with AB=AD and CB=CD. Then ∠B = ∠D, but ∠A ≠ ∠C. It is a quadrilateral with exactly two opposite angles equal, but it is not a parallelogram because its opposite sides are not parallel and equal.

Let the two adjacent angles be 3x and 2x.
Adjacent angles of a parallelogram are supplementary. So,
3x + 2x = 180°
5x = 180° ⇒ x = 36°.
The angles are: 3(36°) = 108° and 2(36°) = 72°.
Since opposite angles are equal, the four angles of the parallelogram are 108°, 72°, 108°, 72°.

Let the adjacent angles be x and x.
Adjacent angles are supplementary: x + x = 180°
2x = 180° ⇒ x = 90°.
So, two angles are 90° each. Since opposite angles are equal, the other two angles are also 90°.
Thus, each angle measures 90° (the figure is a rectangle).

Given HOP is an exterior angle of 70°.
Interior angle ∠POH = 180° - 70° = 110° (Linear pair).
x = 110° (Opposite angles of a parallelogram are equal: ∠E = ∠POH).
Given ∠EHP = 40°.
y = 40° (Alternate interior angles are equal, since EH || PO and HP is transversal).
We know that adjacent angles are supplementary.
∠E + ∠EHO = 180° ⇒ x + (40° + z) = 180°
110° + 40° + z = 180°
150° + z = 180° ⇒ z = 30°.

(i) In parallelogram GUNS: Opposite sides are equal.
So, GU = SN ⇒ 3y - 1 = 26 ⇒ 3y = 27 ⇒ y = 9.
Also, SG = NU ⇒ 3x = 18 ⇒ x = 6.

(ii) In parallelogram RUNS: Diagonals bisect each other.
So, y + 7 = 20 ⇒ y = 13.
And, x + y = 16 ⇒ x + 13 = 16 ⇒ x = 3.

In quadrilateral KLMN, we are given ∠M = 100° and ∠L = 80°.
∠M + ∠L = 100° + 80° = 180°.
Since the sum of interior angles on the same side of the transversal line ML is 180°, the lines NM and KL are parallel to each other.
A quadrilateral with exactly one pair of parallel sides is a trapezium.
Therefore, the figure is a trapezium, and the parallel sides are NM and KL.

(a) False. Rectangles converse (sides may not be all equal) but squares must have all sides equal.

(b) True. A rhombus satisfies all properties of a parallelogram.

(c) True. Squares have properties of both a rhombus (all sides equal) and a rectangle (all angles 90°).

(d) False. Squares satisfy all properties of a parallelogram.

(e) False. A kite does not necessarily have all opposite sides equal or opposite angles equal.

(f) True. A rhombus satisfies all properties of a kite (two distinct pairs of equal length sides, diagonals are perpendicular).

(g) True. A parallelogram has two pairs of parallel sides, which fulfills the requirement of a trapezium (having at least one pair of parallel sides).

(h) True. Since a square is a parallelogram, it is also a trapezium.

(a) Quadrilaterals with four sides of equal length are Square and Rhombus.

(b) Quadrilaterals with four right angles are Square and Rectangle.

(i) A quadrilateral: A square is a simple closed figure bounded by four line segments.

(ii) A parallelogram: Opposite sides of a square are parallel and equal, and opposite angles are equal.

(iii) A rhombus: All four sides of a square are of equal length, and diagonals bisect each other at right angles.

(iv) A rectangle: In a square, all four angles are right angles (90°) and opposite sides are equal.

(i) Bisect each other: Parallelogram, Rhombus, Rectangle, Square.

(ii) Perpendicular bisectors of each other: Rhombus, Square.

(iii) Are equal: Rectangle, Square.

A rectangle is a convex quadrilateral because all of its interior angles are less than 180° (each is 90°), and both of its diagonals lie completely in the interior of the rectangle. No part of either diagonal lies outside the figure.

Extend BO to D such that BO = OD. Join AD and CD.
Since BO = OD and AO = OC (O is midpoint of AC), the diagonals AC and BD bisect each other at O.
So, ABCD is a parallelogram.
But ∠B is 90° (since ABC is a right-angled triangle).
A parallelogram with one right angle is a rectangle.
In a rectangle, diagonals are equal (AC = BD) and bisect each other.
Therefore, OA = OB = OC = OD.
This proves that O is equidistant from A, B, and C.

Number of sides (n) = 12.
Sum of interior angles = (n - 2) × 180°
= (12 - 2) × 180°
= 10 × 180° = 1800°.

An octagon has 8 sides (n = 8).
Sum of all exterior angles = 360°.
Each exterior angle = 360° / n
= 360° / 8 = 45°.

Interior angle = 108°.
Exterior angle = 180° - Interior angle
= 180° - 108° = 72°.
Number of sides (n) = 360° / (Each exterior angle)
= 360° / 72° = 5.
The polygon has 5 sides (it is a pentagon).

Let the fourth angle be x.
Sum of angles of a quadrilateral = 360°.
50° + 80° + 110° + x = 360°
240° + x = 360°
x = 360° - 240° = 120°.

Let the angles be 2x, 3x, 5x, and 8x.
Sum of angles = 360°
2x + 3x + 5x + 8x = 360°
18x = 360°
x = 360° / 18 = 20°.
The angles are:
2x = 2(20°) = 40°
3x = 3(20°) = 60°
5x = 5(20°) = 100°
8x = 8(20°) = 160°.

In a parallelogram, opposite angles are equal.
Therefore, angle C = angle A = 70°.
Adjacent angles are supplementary.
Angle A + Angle B = 180°
70° + Angle B = 180° ⇒ Angle B = 180° - 70° = 110°.
Opposite angles are equal, so Angle D = Angle B = 110°.

Let one side be x cm.
The adjacent side will be (x + 25) cm.
Perimeter = 2(Sum of adjacent sides) = 150
2(x + x + 25) = 150
2x + 25 = 150 / 2 = 75
2x = 75 - 25 = 50
x = 25 cm.
Therefore, the sides are 25 cm and (25 + 25) = 50 cm.

Angle BOC and Angle AOD are vertically opposite angles. So, Angle BOC = 110°.
In triangle AOB, Angle AOB + Angle AOD = 180° (linear pair on diagonal BD).
Angle AOB = 180° - 110° = 70°.
In a rectangle, diagonals are equal and bisect each other, so OA = OB.
This means triangle AOB is isosceles, making Angle OAB = Angle OBA.
Angle OAB + Angle OBA + Angle AOB = 180°
2 × Angle OAB + 70° = 180°
2 × Angle OAB = 110° ⇒ Angle OAB = 55°.

A rhombus is a parallelogram, so opposite angles are equal and adjacent angles are supplementary.
Angle C = Angle A = 60°.
Angle B + Angle A = 180°
Angle B + 60° = 180° ⇒ Angle B = 120°.
Angle D = Angle B = 120°.

Let the adjacent angles be 4x and 5x.
Adjacent angles in a parallelogram are supplementary.
4x + 5x = 180°
9x = 180° ⇒ x = 20°.
The angles are 4(20°) = 80° and 5(20°) = 100°.
Since opposite angles are equal, the four angles are 80°, 100°, 80°, and 100°.

Diagonals of a rhombus bisect each other at right angles (90°).
Therefore, they form 4 right-angled triangles inside the rhombus.
Let the intersection be O, and sides be forming triangle AOB.
OA = half of first diagonal = 16 / 2 = 8 cm.
OB = half of second diagonal = 12 / 2 = 6 cm.
By Pythagoras theorem in triangle AOB:
AB² = OA² + OB²
AB² = 8² + 6² = 64 + 36 = 100.
AB = √100 = 10 cm.
Therefore, the length of each side is 10 cm.

False. A rectangle has opposite sides equal and all angles 90°. A square must have all four sides equal. Thus, not all rectangles are squares.

Formula for number of diagonals in a polygon of n sides is: n(n - 3) / 2.
For a hexagon, n = 6.
Number of diagonals = 6(6 - 3) / 2 = 6(3) / 2 = 18 / 2 = 9 diagonals.

Let the interior angle be x. Then exterior angle = x / 5.
We know, Interior angle + Exterior angle = 180°.
x + (x / 5) = 180°
6x / 5 = 180°
6x = 900° ⇒ x = 150°.
Exterior angle = 150° / 5 = 30°.
Number of sides = 360° / Exterior angle = 360° / 30° = 12 sides.

In a parallelogram, diagonals bisect each other.
Therefore, OA = OC and OB = OD.
Since OA = 4 cm, OC = 4 cm.
Since OB = 5 cm, OD = 5 cm.

The quadrilaterals whose diagonals bisect each other are Parallelogram, Rectangle, Rhombus, and Square.

(c) Diagonals are always equal. This is false. Diagonals are only equal in special parallelograms like rectangles and squares, not in all parallelograms.

Let ABCD be a quadrilateral. Draw a diagonal AC. This divides the quadrilateral into two triangles, ABC and ADC.
In ΔABC, Sum of angles = 180°.
In ΔADC, Sum of angles = 180°.
Sum of angles of quadrilateral ABCD = Sum of angles of ΔABC + Sum of angles of ΔADC
= 180° + 180° = 360°.

Sum of angles = 360°.
x + 2x + 3x + 4x = 360°
10x = 360°
x = 360 / 10 = 36°.

In a parallelogram, opposite angles are equal. Angle A and Angle C are opposite angles. Since 70° is not equal to 65°, No, it cannot be a parallelogram.