# Question 2

Â**In a quadrilateral ABCD, the measure of the three angles A, B and C of the quadrilateral are 110Â°,Â 70Â°Â and 80Â° respectively. Find the measure of the third angle.Â**

**Solution:**

The measure of A = 110Â°

The measure of B = 70Â°

The measure of C = 80Â°

The sum of the four angles of the quadrilateral ABCD = âˆ A + âˆ B + âˆ C +âˆ D=360Â°.

âˆ A + âˆ B + âˆ CÂ = 110Â°+70Â°+80Â° = 260Â°

âˆ A + âˆ B + âˆ C +âˆ D = 360Â°

âˆ D = 360Â°-(âˆ A + âˆ B + âˆ C)

= 360Â°-260Â°

= 100Â°

# Question 3

Â

**In a quadrilateral ABCD, âˆ D is equal to 150Â° and âˆ A = âˆ B = âˆ C.Â Find âˆ A, âˆ B and âˆ C.**

**Solution:**

Measure of âˆ D = 150Â°

Let âˆ A= âˆ B = âˆ C = xÂ°

Sum of the angles of the quadrilateral is 360Â°.

â‡’ xÂ° +xÂ° +xÂ° +150Â° = 360Â°

â‡’ 3xÂ° +150Â° = 360Â°

â‡’ 3xÂ° = 360Â° -150Â° = 210Â°

âˆ´ x = = 70Â°

âˆ´ âˆ A = 70Â°, âˆ B = 70Â° and âˆ C = 70Â°.

# Question 4

Â**The angles of a quadrilateral are in the ratio 1:2:3:4.Â What are the measures of the four angles?**Â Â Â Â Â Â Â Â Â

**Solution:**

Given the ratio of the angles of a quadrilateral = 1:2:3:4

Therefore, let the angles of the quadrilateral be x, 2x, 3x and 4x.

The sum of the angles of a quadrilateral is 360Â°.

â‡’ x+2x+3x+4x = 360Â°

â‡’ 10x = 360Â°

â‡’ x = 36Â°

â‡’ 2x = 2 Ã— 36Â° = 72Â°

â‡’ 3x = 3 Ã— 36Â° = 108Â°

â‡’ 4x = 4 Ã— 36Â° = 144Â°

âˆ´ The measures of the four angles are 36Â°, 72Â°, 108Â° and 144Â°.Â Â Â

# Question 5

Â**The In a quadrilateral**

**(i) which of them have their diagonals bisecting each other?**

(ii) which of them have their diagonals perpendicular to each other?

(iii) which of them have equal diagonals ?

Â

(ii) which of them have their diagonals perpendicular to each other?

(iii) which of them have equal diagonals ?

Â

**Solution:**

Diagonals bisect each other in

a) parallelogram

b) rhombus

c) rectangle

d) Square

e) Kite

(ii) Diagonals are perpendicular in

a) rhombus

b) Square

c) Kite

(iii) Diagonals are equal to each other in

a) rectangle.

b) square

# Question 6

Â

**Adjacent sides of a rectangle are in the ratio 5: 12, if the perimeter of the rectangle is 34cm, findÂ theÂ length ofÂ the diagonal.Â Â Â Â Â Â Â Â Â Â****Solution:**

Given the adjacent sides of a rectangle are in the ratio 5:12.

Therefore let the sides be 5x and 12x.

Then 5x + 12x + 5x + 12x = 34

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 34x = 34

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â x = 1cm

Hence the sides are 12cm and 5cm.

5cm

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 12 cm

The length of the diagonal = âˆš(5^{2} +Â 12^{2}) ) (In a right angled triangle applying Pythagoras theorem)

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = âˆš(25 +Â 144)

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = âˆš169 = 13cm.Â Â Â Â

Therefore the length of the diagonal is 13cm.

# Question 7

Â**The opposite angles of a parallelogram are (3x + 5)**

^{o}and (61 - x)^{o}. Find the measure of fourÂ angles.**Solution:**

(3x + 5)Â =Â (61 - x)Â (Opposite angles of a parallelogram are equal)

Â 3x + x = 61 - 5

Â Â Â Â Â Â Â Â Â 4x = 56

^{o}

x=

Â Â Â Â Â Â Â Â Â Â x = 14^{o}

3x + 5Â =Â 3(14) + 5Â =Â 42 + 5Â =Â 47^{o}

Â 61 - xÂ =Â 61 â€“ 14Â =Â 47^{o}

Angle adjacent to one of the aboveÂ angle = 180^{o} â€“ 47^{o}

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 133^{o}Â (Sum of adjacent angles in a parallelogram is 180^{o})

Fourth angle = 133^{o} (Opposite angles of a parallelogram are equal)

Therefore the four angles in a parallelogram are 47^{o}, 133^{o}, 47^{o} and 133^{o}

# Question 8

Â**ABCD is a ||gm with âˆ A = 80**

^{0}. The internal bisectors of âˆ B and âˆ C meet at O. Find the measure of the three angles of Î” BCO.**Solution:**

âˆ C = âˆ AÂ (Opposite angles of a ||gm are equal)
âˆ C = 800 (Given âˆ C = 80 âˆ OCB = = = 40 |
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â |

âˆ B = 180^{0} - âˆ AÂ Â Â Â (Sum of interior angles on the same side of the transversal is 180^{0 })

Â Â Â Â Â = 180^{0} - 80^{0Â }

Â Â Â Â Â = 100^{0 }

âˆ CBO = = = 5O^{0 }âˆ BOC = 180^{0} â€“ (âˆ OBC + âˆ CBO)Â (Angle sum of a Î” )

Â Â Â Â Â Â Â Â Â Â = 180^{0} â€“ (40^{0} + 50^{0})

Â Â Â Â Â Â Â Â Â Â = 180^{0} - 90^{0 }

Â Â Â Â Â Â Â Â Â Â = 90^{0 }

**âˆ´ **The Three angles of the triangle BCO namely âˆ OCB, âˆ CBO, âˆ BOC are 40^{0}, 50^{0} and 900 respectively.

# Question 9

Â**Find the measure of all four angles of a parallelogram whose consecutive angles are in the ratio 1 : 3.**

**Solution:**

Given consecutive angles of a parallelogram are in the ratio 1:3

Therefore, the two consecutive angles be x and 3x.

x + 3x = 180

^{0}Â Â Â Â Â Â Â (sum of the interior angles on the same side of the transversal is 180

^{0})

Â Â Â Â Â 4x = 180^{0 }

Â Â Â Â Â Â x = 45^{0 }

Therefore the two consecutive angles are 45^{0 }and 3(45^{0}) = 135^{0}.

Since the opposite angles of a parallelogram are equal. The measures of all four angles of a parallelogram are 45^{0}, 45^{0}, 135^{0} and 135^{0}.

# Question 10

**A diagonal and a side of a rhombus are of equal length. Find the measure of the angles of the rhombus.**

**Solution:**

Let ABCD be the rhombus. AB = BC = DC = DA (sides of a rhombus are equal) But AB = BD (Given) âˆ´ AB = BC = CD = DA = BD Since in Î” ABD all the sides are equal. Î” ABD is an equilateral Î” . Similarly Î” BCD is also an equilateral. |

\ ∠ A = âˆ ABD = âˆ ADB = âˆ DBC = âˆ C = âˆ CDB = 60^{0 }

âˆ´ âˆ B = âˆ ABD + âˆ DDC = 60^{0} + 60^{0} = 120^{0 }

and âˆ D = âˆ ADB + âˆ CDB = 60^{0} + 60^{0} = 120^{0}

âˆ´ The angles of the rhombus are 60^{0}, 120^{0}, 60^{0} and 120^{0}.