Skip to main content

3

Q1. The measure of two angles of a quadrilateral are 115o and 45o , and the other two angles are equal. Find the measure of each of the equal angles.

Solution

Let the measure of the each equal angle be xo ∴ 115 + 45 + x + x = 360 (As the sum of angles of quadrilateral is 360o) ⇒ 160 + 2x = 360 ⇒ 2x = 360 - 160  ⇒ x = 100 Hence, the measure of each of the equal angles is 100o .
Q2. Find the sum of the angles of a polygon with 13 sides. 

Solution

Sum space of space angles space of space straight a space polygon equals left parenthesis straight n space minus space 2 right parenthesis space cross times space 180 degree rightwards double arrow Sum space of space angles space of space straight a space polygon space with space 13 space sides space space space space space space equals space left parenthesis 13 space – space 2 right parenthesis space cross times space 180 degree space space space space space space space equals space 11 space cross times space 180 degree space space space space space space space equals space 1980 degree space space Thus comma space the space sum space of space the space angles space of space straight a space given space polygon space is space 1980 degree. space
Q3. The measure of interior angle of a regular polygon is eight times the measure of its exterior angle. Find the number of sides in the polygon.

Solution

Let space the space measure space of space an space exterior space angle space be space straight x. Then comma space the space measure space of space an space interior space angle space equals space 8 straight x We space know space that space the space interior space and space exterior space angle space form space straight a space linear space pair. rightwards double arrow straight x space plus space 8 straight x space equals space 180 degree rightwards double arrow 9 straight x space equals space 180 degree rightwards double arrow straight x space equals space 180 degree space divided by space 9 space equals space 20 degree space space Sum space of space exterior space angles space of space straight a space polygon space equals space 360 degree space rightwards double arrow Number space of space sides space of space polygon space equals fraction numerator 360 degree over denominator 20 degree end fraction equals 18 space Hence comma space the space number space of space sides space in space the space polygon space is space 18. space
Q4. PQRS is a rectangle whose diagonals meet at O. If OQ = x + 5 and OR = 2x + 3, then find x. 

Solution

Since space diagonals space of space straight a space rectangle space are space equal space and space bisect space each space other. therefore space OQ equals 1 half SQ space and space OR equals 1 half PR Also comma space SQ equals PR rightwards double arrow 1 half SQ equals 1 half PR rightwards double arrow space straight x space plus space 5 equals 2 straight x space plus space 3 rightwards double arrow straight x equals 2 space
Q5. Three angles of a quadrilateral are in the ratio 4 : 6 : 5, if the measure of fourth angle is 60°, find the angles.

Solution

Let space the space measure space of space three space angles space be space 4 straight x comma space 6 straight x space and space 5 straight x. Fourth space angle space equals space 60 degree space space space space The space sum space of space interior space angles space of space straight a space quadrilateral space is space 360 degree. rightwards double arrow 4 straight x space plus space 6 straight x space plus space 5 straight x space plus space 60 degree space equals space 360 degree rightwards double arrow 15 straight x space plus space 60 degree space equals space 360 degree rightwards double arrow 15 straight x space equals space 300 degree rightwards double arrow straight x space equals space fraction numerator 300 degree over denominator 15 end fraction equals 20 degree rightwards double arrow 4 straight x equals 4 cross times 20 degree equals 80 degree comma space space 6 straight x equals 6 cross times 20 degree equals 120 degree space and space 5 straight x equals 5 cross times 20 degree equals 100 degree Hence comma space the space other space three space angles space are space 80 degree comma space 120 degree space and space 100 degree. space
Q6. Two angles of a quadrilateral are 45° and 75°. If the measure of other two angles are in the ratio 5 : 7, then find the measure of each angle.

Solution

Let space the space measure space of space other space two space angles space be space 5 straight x space and space 7 straight x space respectively. Now comma space the space sum space of space interior space angles space of space straight a space quadrilateral space is space 360 degree. rightwards double arrow 45 degree space plus space 75 degree space plus space 5 straight x space plus space 7 straight x space equals space 360 degree rightwards double arrow 120 degree space plus space 12 straight x space equals space 360 degree rightwards double arrow 12 straight x space equals space 240 degree rightwards double arrow space straight x space equals space 20 degree rightwards double arrow space 5 straight x space equals space 5 space cross times space 20 degree space equals space 100 degree space and space 7 straight x space equals space 7 space cross times space 20 degree space equals space 140 degree Thus comma space the space measure space of space other space two space angles space are space 100 degree space and space 140 degree.
Q7. The perimeter of a parallelogram is 170 cm. One of its sides is greater than the other by 15 cm. Find the lengths of all the sides of the parallelogram.

Solution

Let space one space side space of space the space parallelogram space equals space space straight x space cm Then comma space its space other space side space equals space left parenthesis straight x plus 15 right parenthesis space cm Now comma space opposite space sides space of space straight a space paralleogram space are space equal. Hence comma space its space remaining space are space straight x space cm space and space left parenthesis straight x plus 15 right parenthesis space cm. Perimeter space of space straight a space parallelogram equals straight x space plus space left parenthesis straight x plus 15 right parenthesis space plus space straight x space plus space left parenthesis straight x plus 15 right parenthesis But space perimeter equals 170 space cm space space left parenthesis given right parenthesis rightwards double arrow 4 straight x space plus space 30 space equals space 170 rightwards double arrow 4 straight x space equals space 140 rightwards double arrow straight x equals space 35 space cm rightwards double arrow straight x plus 15 space equals space 35 plus 15 space equals space 50 space cm Hence comma space the space lengths space of space all space sides space of space straight a space parallelogram space are space 35 space cm comma space 50 space am comma space 35 space cm space and space 50 space cm.
Q8. A quadrilateral has three interior angles each equal to 75°. Find the measure of the fourth interior angle.

Solution

Let space the space fourth space inetrior space angle space be space straight x. The space sum space of space the space interior space angles space of space straight a space quadrilateral space is space 360 degree. rightwards double arrow straight x space plus space 75 degree plus 75 degree plus 75 degree equals 360 degree rightwards double arrow straight x space plus space 3 cross times 75 degree equals 360 degree rightwards double arrow straight x space plus space 225 degree equals 360 degree rightwards double arrow straight x space equals space 360 degree space minus space 225 degree space equals 135 degree Hence comma space the space measure space of space the space fourth space interior space angle space is space 135 degree.
Q9. If the angles of the septagon are in the ratio 1 : 5 : 3 : 5 : 1 : 1 : 2, then calculate the measure of these angles.

Solution

Let space the space angles space of space the space septagon space be space straight x comma space 5 straight x comma space 3 straight x comma space 5 straight x comma space straight x comma space straight x space and space 2 straight x. Now comma straight x space plus space 5 straight x space plus space 3 straight x space plus space 5 straight x space plus space straight x space plus straight x space plus 2 straight x space equals space left parenthesis 7 space minus space 2 right parenthesis space cross times 180 degree rightwards double arrow 18 straight x space equals space 5 space cross times space 180 degree rightwards double arrow space straight x space equals space fraction numerator 5 space cross times space 180 degree over denominator 18 end fraction equals 50 degree rightwards double arrow 5 straight x space equals space 5 space cross times space 50 degree space equals space 250 degree space space space space space space 3 straight x space equals space 3 space cross times space 50 space equals space 150 degree space space space space space 2 straight x space equals space 2 space cross times space 50 space equals space 100 degree Hence comma space the space angles space are space 50 degree comma space 250 degree comma space 150 degree comma space 250 degree comma space 50 degree comma space 50 degree space and space 100 degree. space
Q10. Is it possible to have a regular polygon whose each interior angle is 150°?

Solution

Each space interior space angle equals 150 degree space space space space left parenthesis Given right parenthesis therefore Each space exterior space angle equals 180 degree minus 150 degree equals 30 degree rightwards double arrow space Number space of space sides space of space the space regular space polygon equals fraction numerator 360 degree over denominator 30 degree end fraction equals 12 degree comma space which space is space straight a space natural space number. Hence comma space it space is space possible space to space have space such space straight a space regular space polygon.


Comments

Post a Comment