Question

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Answer 1

1. Intersecting lines are lines that cross each other. When two lines intersect, the opposite angles are equal.

Given: x + 30° and 180° - 2x are vertically opposite angles

⇒ x + 30° = 180° - 2x

⇒ 3x = 150°

⇒ x = 50°

2. Let the complement be : C

Given : The other angle is two third of its complement

`\sf{\implies Other \ angle = (2)/(3) * C}`

Two angles are said to be Complementary if their sum is equal to 90°

`\sf{\implies C + (2C)/(3) = 90}`

`\sf{\implies (5C)/(3) = 90}`

`\sf{\implies 5C = 270}`

`\sf{\implies C = 54}`

`\sf{\implies Other \ angle = \bigg((2)/(3) * C\bigg) = \bigg((2)/(3) * 54\bigg) = 36 }`

Answer: The size of the angle which is two third of its complement is 36°

3. Let the supplement be : S

Given : The other angle is one fifth of its supplement

`\sf{\implies Other \ angle = (1)/(5) * S}`

Two angles are said to be Supplementary if their sum is equal to 180°

`\sf{\implies S + (S)/(5) = 180}`

`\sf{\implies (6S)/(5) = 180}`

`\sf{\implies 6S = 900}`

`\sf{\implies S = 150}`

`\sf{\implies Other \ angle = \bigg((1)/(5) * S\bigg) = \bigg((1)/(5) * 150\bigg) = 30 }`

Answer: The size of the angle which is one fifth of its supplement is 30°

4. Two supplementary angles are in the ratio 7 : 3

Let the supplementary angles be : 7S and 3S

Two angles are said to be Supplementary if their sum is equal to 180°

⇒ 7S + 3S = 180°

⇒ 10S = 180°

⇒ S = 18°

First Angle : 7S = (7 × 18) = 126°

Second Angle : 3S = (3 × 18) = 54°

Answer 2

Explanation:

1) Vertically opposite angles are equal.

x + 30 = 180 - 2x

Add '2x' to both sides

x + 30 +2x = 180

3x + 30 = 180

Subtract 30 from both sides

3x = 180 - 30

3x = 150

x = 150/3

x = 50

2) Let the angle be x

It's complement angle = 90 - x

`x = (2)/(3)(90-x)\\\\x = (2)/(3)*90 - (2)/(3)*x\\\\x =2*30 - (2)/(3)x\\\\x +(2)/(3)x = 60\\\\(3)/(3)x+(2)/(3)x=60\\\\(5)/(3)x =60\\\\x = 60*(3)/(5)\\\\x = 12*3\\x = 36`

The angles are : 36 , 54

3) Let the angle be x

It's supplement = 180 - x

`x = (1)/(5)(180-x)\\\\x =(1)/(5)*180 - (1)/(5)x\\\\x + (1)/(5)x= 36\\\\(5)/(5)x +(1)/(5)x = 36\\\\(6)/(5)x = 36\\\\x = 36*(5)/(6)\\\\x = 6*5 = 30\\`

Angles are 30 , 150

4) Ratio = 7 : 3

The angles are = 7x , 3x

7x + 3x = 180

10x = 180

x = 180/10

x = 18

7x = 7 * 18 = 126

3x = 3 * 18 = 54

Angles are 126 , 54