1620°

The sum of the measures of all angles of a convex quadrilateral is 360°. Yes, this property holds, if the quadrilateral is not convex.
e.g. ABCD is a concave quadrilateral, but sum of the measures of its angles is 360°.

We have, n=18
Each interior angle = \(\frac { { 180 }^{ 0 }\times (18-2) }{ 18 } =\frac { { 180 }^{ 0 }\times 16 }{ 18 } \)=160⁰.

We have, n=12
Each interior angle = \(\frac { { 180 }^{ 0 }\times (n-2) }{ n } =\frac { { 180 }^{ 0 }\times (12-2) }{ 12 } =\frac { { 1800 }^{ 0 } }{ 12 } \)=150⁰

∵ Number of sides in hexagon (n) =6
∴ Number of diagonals =\(\frac { n(n-3) }{ 2 } =\frac { 6(6-3) }{ 2 } =\frac { 18 }{ 2 } \)=9

(a) We know that, the sum of the measure of the exrernal angles of any polygon is 360°.
Here, exterior angles are x, y. z, p, q and r.
 ∴ x + y + z + P + q + r = 360°
(b) Yes, x = y = z = p = q = r, because each of them is equal to 180° - ㄥa.
(c) (i) Here, all exterior angles are equal and their sum is 360°, so the measure of each exterior angle is x=\(\frac { 360^{ 0 } }{ 6 } \)=60⁰.
(ii) All interior angles are also equal to each other, where interior angle is a = 180° - r [by linear pair]
∴ Measure of each interior angle = 180° - 60° =120°
(d) (i) The regular octagon having 8 equal sides.
∴ All the exterior angles have equal measure, say x.
Then, 8x = 360°
Since, sum of the measures of the external angles of any polygon is 360°.
x = \(\frac { 360^{ 0 } }{ 8 } \) = 45⁰.
Therefore, measure of each exterior angle = 45° and measure of each interior angle = 180 ° - 45° =135°.
(ii) The regular 20-gon polygon has 20 equal sides.
∴ All the exterior angles have equal measure, say x. Then, 20x = 360°
Since, sum of the measures of the external angles of any polygon is 360°.
∴ x = \(\frac { 360^{ 0 } }{ 20 } \)=1 8⁰
Therefore, measure of each exterior angle = 18°
and measure of each interior angle = 180° -18° =162°

(a) Simple curve A plane figure formed by joining a number of points without lifting a pencil from the paper and without returing any portion of the drawing other than single points is called a simple curve or curve. In the given figures, simple curves are figures (i), (ii), (v), (vi) and (vii).
(b) Simple closed curve A closed curve, which does not intersect itself, is called a simple closed curve. In the given figures, simple closed curves are figures (i), (ii), (v), (vi) and (vii).
(c) Polygon A polygon is a closed curve formed by the line segments such that
(i) no two line segments intersect except at their end points.
(ii) no two line segments with a common end points are coincide. In other words, a simple dosed curve made upto only line segments is called a polygon. In the given figures, polygons are figures (i) and (ii).
(d) Convex polygon A convex polygon is a polygon in which each interior angle has a measure less than 180°. In other words, a polygon is convex, if noportion of their diagonals in their exterior. In the given figures, convex polygon is figure (ii).
(e) Concave polygon A concave polygon is a polygon, which atleast one interior angle has measure more than 180°, i.e. atleast one segment connecting two vertices is outside the polygon. In the given figure, concave polygons are figures (i) and (iv).

Let ABCD be a parallelogram in which two adjacent angles
ㄥA and ㄥB have equal measure, say x .
Then, mㄥA = x and mㄥB = x
We know that, two adjacent angles in a parallelogram are supplementary.
mㄥA + mㄥB = 180° ⇒ x + x =180° ⇒  2x =180°
⇒ x = \(\frac { 180^{ 0 } }{ 2 } \)= 90⁰
So, mㄥA = 90° and mㄥB = 90°
Also, the opposite angles of a parallelogram are of equal measure.
Therefore, mㄥC = mㄥA = 90° and mㄥD = mㄥB = 90°.

Let two adjacent angles A and B of a parallelogram ABCD be 3x and 2x, respectively.
Weknow that, the sum of two
adjacent tangles of a parallelogram is 180⁰
∴ ㄥA+ㄥB=180⁰
⇒ 3x + 2x = 180⁰
⇒ 5x =180°  ⇒ \(\frac { 180^{ 0 } }{ 5 } \)= 360
Therefore, ㄥA = 3x = 3 x 36⁰ = 108⁰
and ㄥB = 2x = 2 x 36° = 72°
Also, we know that, opposite angles of a parallelogram are of equal measure.
∴ ㄥC = ㄥA =108° and ㄥD = ㄥB = 72°
Hence, the angles of a parallelogram are ㄥA = 108°,
ㄥB = 72°, ㄥC = 108° and ㄥD = 72°, respectively.

Let given polygon be
ABCDE It is clear that, AB is a straight line.
90° + ㄥ1 = 180°
[by linear pair angle]
⇒ ㄥ1 = 180⁰ - 90⁰ = 90⁰
Now, x +90° +60° +90° + 70° = 360°
The sum of the exterior angles of any polygon is 360°.
⇒ x + 310⁰ = 360°
⇒ x = 360⁰-310⁰=  50°
Hence, the measure of angle x is 50°.

We know that, the sum of three angles of a triangle is 180°.

Let the given triangle be ABC.
∴ ㄥA + ㄥB+ ㄥC =180°
⇒  30⁰ + 90⁰+ ㄥC=180° ⇒  ㄥC+120⁰=180°
⇒  ㄥC = 180° - 120° = 60°
Now, ㄥEBC + ㄥABC = 180° [by linear pair]
∴  x +90° = 180° ⇒ x = 180° - 90° = 90°
Similarly, ㄥFCA + ㄥACB = 180° [by linear pair]
⇒ y+60⁰=180° ⇒ y=180⁰-60⁰=120°
and ㄥDAB + ㄥBAC = 180° [by linear pair]
⇒ z + 30° = 180° ⇒ z = 180° - 30° = 150°
∴ x + y + z = 90° + 120° + 150° = 360°
Hence, the value of x + y + z is 360°

From the given table, we observe that the sum of angles
(interior angles) of a polygon having m sides is (m - 2) x 180°.
(a) Here, number of sides of a polygon (m) = 7
∴ Sum of the angles of a polygon of 7 sides
= (7 - 2) x 180° = 5 x 180° = 900°
(b) Here, number of sides of a polygon (m) = 8 .
∴ Sum of the angles of a polygon of 8 sides
= (8 - 2) x 180° = 6 x 180° = 1080°
(c) Here, number of sides of a polygon (m) = 10
∴ Sum of the angles of a polygon of 10 sides
= (10 - 2) x 180° = 8 Xx 180° = 1440°
(d) Here, number of sides of a polygon (m) = n
∴ Sum of the angles of a polygon of n sides
= (n-2) x 180°.

Supplementary; adjacent angles of a parallelogram are supplementary.

8 cm; [∵ in a rectangle, diagonals are of equal length]

Parallelogram; since, in a parallelogram, diagonals bisect each other.

An equilateral triangle; since, in an equilateral triangle, all the sides and angles are equal.

Square; since, in a square all the sides and angles are equal.