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
[tex]\sf{\implies Other \ angle = \dfrac{2}{3} \times C}[/tex]
Two angles are said to be Complementary if their sum is equal to 90°
[tex]\sf{\implies C + \dfrac{2C}{3} = 90}[/tex]
[tex]\sf{\implies \dfrac{5C}{3} = 90}[/tex]
[tex]\sf{\implies 5C = 270}[/tex]
[tex]\sf{\implies C = 54}[/tex]
[tex]\sf{\implies Other \ angle = \bigg(\dfrac{2}{3} \times C\bigg) = \bigg(\dfrac{2}{3} \times 54\bigg) = 36 }[/tex]
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
[tex]\sf{\implies Other \ angle = \dfrac{1}{5} \times S}[/tex]
Two angles are said to be Supplementary if their sum is equal to 180°
[tex]\sf{\implies S + \dfrac{S}{5} = 180}[/tex]
[tex]\sf{\implies \dfrac{6S}{5} = 180}[/tex]
[tex]\sf{\implies 6S = 900}[/tex]
[tex]\sf{\implies S = 150}[/tex]
[tex]\sf{\implies Other \ angle = \bigg(\dfrac{1}{5} \times S\bigg) = \bigg(\dfrac{1}{5} \times 150\bigg) = 30 }[/tex]
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°
