Question

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Answer 1

`\textsf {1. x = 28}`

`\textsf {2. a = c = e = f = 55 and b = d = 125}`

Explanation:

`\textsf {Question 1}`

`\textsf {Here, the Angle Sum Property needs to be remembered, }\\\textsf {which states that the internal angle sum of a triangle is }\\\textsf {equal to 180 degrees.}`

`\textsf {Solving :}`

`\implies \mathsf {5x - 60 + 2x + 40 + 3x - 80 = 180}`

`\implies \mathsf {10x - 100 = 180}`

`\implies \mathsf {10x = 280}`

`\implies \mathsf {x = (280)/(10)}`

`\implies \mathsf {x = 28}`

`\textsf {Question 2}`

`\textsf {Now, remember the sum of linear angles is 180,}\\ \textsf{vertical angles are equal, and corresponding angles are equal.}`

`\textsf {Hence, e = f = a = c = 180 - 125}`

`\implies \mathsf {a = c = e = f = 55}`

`\textsf {Also, b and d are equal to the listed angle as}\\ \textsf{d corresponds to it, and b is the vertical angle of d. }`

`\implies \mathsf {b = d = 125}`

Answer 2

`\star\:{\underline{\underline{\sf{\purple{ \: Question \: 1\: }}}}}`

`{\large{\textsf{\textbf{\underline{\underline{Given \: :}}}}}}`

‣ Angle A = 5x - 60°

‣ Angle B = 2x + 40°

‣ Angle C = 3x - 80°

`{\large{\textsf{\textbf{\underline{\underline{To \: Find \: :}}}}}}`

‣ The value of
`x`

`{\large{\textsf{\textbf{\underline{\underline{Solution \: :}}}}}}`

By angle sum property [ASP] of a triangle which states that the sum of all angles of a triangle = 180°

`\longrightarrow \tt A+B+C =180°`

`\longrightarrow \tt (5x - 60) + (2x + 40) + (3x - 80) =180`

`\longrightarrow \tt 5x + 2x + 3x - 60 + 40 - 80 =180`

`\longrightarrow \tt 10x - 60 + 40 - 80 =180`

`\longrightarrow \tt 10x - 140 + 40 =180`

`\longrightarrow \tt 10x - 100 = 180`

`\longrightarrow \tt 10x = 180 + 100`

`\longrightarrow \tt x = \cancel{(280)/(10) }`

`\longrightarrow \tt x = \purple{28 \degree}`

Therefore, the value of
`x` is 28°

`\star\:{\underline{\underline{\sf{\red{ \: Question \: 2\: }}}}}`

`{\large{\textsf{\textbf{\underline{\underline{Given \: :}}}}}}`

‣ Line p is parallel to line q which is intersected by a transversal.

`{\large{\textsf{\textbf{\underline{\underline{To \: Find \: :}}}}}}`

‣ The unknown angles.

`{\large{\textsf{\textbf{\underline{\underline{Solution \: :}}}}}}`

Finding angle
`e`

[linear pair axiom]

`\longrightarrow \tt 125 \degree + \angle e = 180 \degree`

`\longrightarrow \tt \angle e = 180 \degree - 125 \degree`

`\longrightarrow \tt \angle e = \red{55 \degree}`

Now,

For angle
`f`

[Vertically opposite angles]

`\longrightarrow \tt \angle f = \angle e`

`\longrightarrow \tt \angle f = \green{55 \degree }`

Now,

For angle
`a`

[Corresponding angles]

`\longrightarrow \tt \angle a = \angle e`

`\longrightarrow \tt \angle a = \orange{55 \degree}`

Now,

For angle
`d`

[Corresponding angles]

`\longrightarrow \tt \angle d = \pink{125 \degree}`

Now,

For angle
`c`

[Vertically opposite angles]

`\longrightarrow \tt \angle c = \angle a`

`\longrightarrow \tt \angle c = \gray{ 55 \degree}`

Now,

For angle
`b`

[Vertically opposite angles]

`\longrightarrow \tt \angle b = \angle d`

`\longrightarrow \tt \angle b = \purple{ 125 \degree}`

Hence,

★ Angle A = 55°

★ Angle B = 125°

★ Angle C = 55°

★ Angle D = 125°

★ Angle E = 55°

★ Angle F = 55°

`{\underline{\rule{290pt}{2pt}}}`