## Polygons+Quadrilateral+Parallelogram

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*Polygons+Quadrilateral+Parallelogram*

*Polygons+Quadrilateral+Parallelogram*

*1. How many diagonals does each of the following have?*

(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

*2. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? Using the formula: (n - 2)180°*

*3.What can you say about the angle sum of a convex polygon with number of sides?(a) 7 (b) 8 (c) 10*

*4. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides*

*5. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides*

*8. How many sides does a regular polygon have if the measure of an exterior angle is 24°? Number of sides of a polygon =360/exterior angle*

*9. How many sides does a regular polygon have if each of its interior angles is 165°? Exterior Angle = 180°-interior angle*

*10. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?*

(b) Can it be an interior angle of a regular polygon? Why?(b) Can it be an interior angle of a regular polygon? Why?

*[ If interior angle is 22° then the exterior angle = 180°-22°=158°On dividing 360° by 158° we can’t get answer in whole number, so such a polygon is not possible.]*

*11. (a) What is the minimum interior angle possible for a regular polygon? Why?*

(b) What is the maximum exterior angle possible for a regular polygon?

[Answer: The polygon with minimum number of sides is a triangle, and each angle of an equilateral triangle measures 60°, so 60° is the minimum value of the possible interior angle for a regular polygon. For an equilateral triangle the exterior angle is 180°-60°=120° and this is the maximum possible value of an exterior angle for a regular polygon.](b) What is the maximum exterior angle possible for a regular polygon?

[Answer: The polygon with minimum number of sides is a triangle, and each angle of an equilateral triangle measures 60°, so 60° is the minimum value of the possible interior angle for a regular polygon. For an equilateral triangle the exterior angle is 180°-60°=120° and this is the maximum possible value of an exterior angle for a regular polygon.]

*12. Can a quadrilateral ABCD be a parallelogram if*

(i) <D + <B=180? (ii) AB=DC= 8cm, AD= 4cm,and BC = 4.4 cm (iii) <A = 70 and <C = 65?

Answer:(i) <D + <B=180? (ii) AB=DC= 8cm, AD= 4cm,and BC = 4.4 cm (iii) <A = 70 and <C = 65?

Answer:

*(i)It can be , but not always as you need to look for other criteria as well.*

(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram.

(iii) Here opposite angles are not equal, so it is not a parallelogram.

(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram.

(iii) Here opposite angles are not equal, so it is not a parallelogram.

*13. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.*

*14. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.*

*15. State whether True or False.*

a) All rectangles are squares

Answer: All squares are rectangles but all rectangles can’t be squares, so this statement is false.

(b) All kites are rhombuses.

Answer: All rhombuses are kites but all kites can’t be rhombus

(c) All rhombuses are parallelograms

Answer: True

(d) All rhombuses are kites.

Answer: True

(e) All squares are rhombuses and also rectangles

Answer: True; squares fulfils all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfil all criteria of a rhombus, as all sides are equal and their diagonals bisect each other.

(f) All parallelograms are trapeziums.

Answer: False;

All trapeziums are parallelograms, but all parallelograms can’t be trapezoid.

(g) All squares are not parallelograms.

Answer: False; all squares are parallelograms

(h) All squares are trapeziums.

Answer: True

all four sides are equal then ita) All rectangles are squares

Answer: All squares are rectangles but all rectangles can’t be squares, so this statement is false.

(b) All kites are rhombuses.

Answer: All rhombuses are kites but all kites can’t be rhombus

(c) All rhombuses are parallelograms

Answer: True

(d) All rhombuses are kites.

Answer: True

(e) All squares are rhombuses and also rectangles

Answer: True; squares fulfils all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfil all criteria of a rhombus, as all sides are equal and their diagonals bisect each other.

(f) All parallelograms are trapeziums.

Answer: False;

All trapeziums are parallelograms, but all parallelograms can’t be trapezoid.

(g) All squares are not parallelograms.

Answer: False; all squares are parallelograms

(h) All squares are trapeziums.

Answer: True

all four sides are equal then it

*16.Identify all the quadrilaterals that have.*

*(a) four sides of equal length (b) four right angles*

Answer:Answer:

*(a) If*

*can be either a square or a rhombus.*

(b) All four right angles make it either a rectangle or a square.(b) All four right angles make it either a rectangle or a square.

*17.Explain how a square is.*

(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle

Answer: (i) Having four sides makes it a quadrilateral

(ii) Opposite sides are parallel so it is a parallelogram

(iii) Diagonals bisect each other so it is a rhombus

(iv) Opposite sides are equal and angles are right angles so it is a rectangle.(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle

Answer: (i) Having four sides makes it a quadrilateral

(ii) Opposite sides are parallel so it is a parallelogram

(iii) Diagonals bisect each other so it is a rhombus

(iv) Opposite sides are equal and angles are right angles so it is a rectangle.

*18. Name the quadrilaterals whose diagonals.*

(i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal

Answer: Rhombus; because, in a square or rectangle diagonals don’t intersect at right angles.(i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal

Answer: Rhombus; because, in a square or rectangle diagonals don’t intersect at right angles.

*19 Explain why a rectangle is a convex quadrilateral.*

Answer: Both diagonals lie in its interior, so it is a convex quadrilateral.Answer: Both diagonals lie in its interior, so it is a convex quadrilateral.

*20 ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C.*

Answer: If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diagonals of the rectangle. In a rectangle diagonals are equal and bisect each other.

So, AC = BD , AO = OC ,BO = OD, And AO = OC = BO = OD So, it is clear that O is equidistant from A, B and C.Answer: If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diagonals of the rectangle. In a rectangle diagonals are equal and bisect each other.

So, AC = BD , AO = OC ,BO = OD, And AO = OC = BO = OD So, it is clear that O is equidistant from A, B and C.

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