1. Find the acute angle between the lines `x+3y+1=0` and `2x-y+4=0`

Solution:
We shall first find slopes of both the lines.

1. Slope of line `x+3y+1=0`

`x+3y+1=0`

`:. 3y=-x-1`

`:. y=-(x)/(3)-1/3`

`:. m_1=-1/3`

2. Slope of line `2x-y+4=0`

`2x-y+4=0`

`:. y=2x+4`

`:. m_2=2`


Let `theta` be the acute angle between the lines

then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`

`=|(-1/3-2)/(1+(-1/3)(2))|`

`=|(-7/3)/(1+-2/3)|`

`=|(-7/3)/(1/3)|`

`=7`

`:. theta=81.87^circ`

Hence, the acute angle between the given lines is `81.87^circ`

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2. Find the acute angle between the lines `3x+2y+4=0` and `2x-3y-7=0`

Solution:
We shall first find slopes of both the lines.

1. Slope of line `3x+2y+4=0`

`3x+2y+4=0`

`:. 2y=-3x-4`

`:. y=-(3x)/(2)-2`

`:. m_1=-3/2`

2. Slope of line `2x-3y-7=0`

`2x-3y-7=0`

`:. 3y=2x-7`

`:. y=(2x)/(3)-7/3`

`:. m_2=2/3`


Let `theta` be the acute angle between the lines

then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`

`=|(-3/2-2/3)/(1+(-3/2)(2/3))|`

`=|(-13/6)/(1+-1)|`

`=|(-13/6)/(0)|`

`=1/0`

`:. theta=90^circ`

Hence, the acute angle between the given lines is `90^circ`

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3. Find the acute angle between the lines `2x+3y+5=0` and `x-2y-4=0`

Solution:
We shall first find slopes of both the lines.

1. Slope of line `2x+3y+5=0`

`2x+3y+5=0`

`:. 3y=-2x-5`

`:. y=-(2x)/(3)-5/3`

`:. m_1=-2/3`

2. Slope of line `x-2y-4=0`

`x-2y-4=0`

`:. 2y=x-4`

`:. y=(x)/(2)-2`

`:. m_2=1/2`


Let `theta` be the acute angle between the lines

then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`

`=|(-2/3-1/2)/(1+(-2/3)(1/2))|`

`=|(-7/6)/(1+-1/3)|`

`=|(-7/6)/(2/3)|`

`=7/4`

`:. theta=60.26^circ`

Hence, the acute angle between the given lines is `60.26^circ`

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4. Find the acute angle between the lines `3x-y+4=0` and `2x+y=3`

Solution:
We shall first find slopes of both the lines.

1. Slope of line `3x-y+4=0`

`3x-y+4=0`

`:. y=3x+4`

`:. m_1=3`

2. Slope of line `2x+y=3`

`2x+y=3`

`:. y=-2x+3`

`:. m_2=-2`


Let `theta` be the acute angle between the lines

then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`

`=|(3--2)/(1+(3)(-2))|`

`=|(5)/(1+-6)|`

`=|(5)/(-5)|`

`=1`

`:. theta=45^circ`

Hence, the acute angle between the given lines is `45^circ`

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5. Find the acute angle between the lines `2x-y+3=0` and `x-3y+7=0`

Solution:
We shall first find slopes of both the lines.

1. Slope of line `2x-y+3=0`

`2x-y+3=0`

`:. y=2x+3`

`:. m_1=2`

2. Slope of line `x-3y+7=0`

`x-3y+7=0`

`:. 3y=x+7`

`:. y=(x)/(3)+7/3`

`:. m_2=1/3`


Let `theta` be the acute angle between the lines

then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`

`=|(2-1/3)/(1+(2)(1/3))|`

`=|(5/3)/(1+2/3)|`

`=|(5/3)/(5/3)|`

`=1`

`:. theta=45^circ`

Hence, the acute angle between the given lines is `45^circ`

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