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In this lesson, we will learn how to find the measure of the acute angle between two straight lines in the plane.

Q1:

Determine the measure of the acute angle between the straight line π₯ β π¦ + 4 = 0 and the straight line passing through the points ( 3 , β 2 ) and ( β 2 , 4 ) to the nearest second.

Q2:

Determine the measure of the positive angle that the straight line πΏ makes with the positive direction of the π₯ -axis approximated to the nearest second, given that πΏ passes through points π΄ ( β 1 , β 4 ) and π΅ ( 3 , β 5 ) .

Q3:

If the acute angle between the straight lines whose equations are π π¦ β 2 π₯ + 1 9 = 0 and 9 π₯ β 7 π¦ β 8 = 0 is π 4 , find all the possible values of π .

Q4:

Given that π΄ π΅ πΆ is a right-angled triangle at π΄ , the equation of β ο© ο© ο© ο© β π΅ πΆ is r = β¨ 1 , 4 β© + πΎ β¨ β 6 , β 4 β© , and the equation of β ο© ο© ο© ο© β π΄ π΅ is r = β¨ 4 , β 9 β© + πΎ β¨ 1 , β 8 β© , find π β π΄ πΆ π΅ approximated to the nearest minute.

Q5:

Determine, to the nearest second, the measure of the positive angle that the straight line 1 4 π₯ + 1 2 π¦ = β 2 7 makes with the positive π₯ -axis.

Q6:

A line passing through ( 8 , 2 ) makes an angle π with the line 6 π₯ + 4 π¦ + 9 = 0 , and t a n π = 1 5 1 3 . What is the equation of this line?

Q7:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 5 and 1 4 .

Q8:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 7 and β 8 7 .

Q9:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of β 1 3 and β 7 .

Q10:

Find the measure of the acute angle between the two straight lines whose equations are 1 1 π₯ + 1 0 π¦ β 2 8 = 0 and 2 π₯ + π¦ + 1 5 = 0 to the nearest second.

Q11:

Find the measure of the acute angle between the two straight lines whose equations are π₯ β 2 π¦ + 2 = 0 and 7 π₯ β 1 0 π¦ + 4 5 = 0 to the nearest second.

Q12:

Determine the measure of the acute angle between the two straight lines πΏ βΆ β π = ( β 4 , β 3 ) + πΎ ( 4 , β 9 ) 1 and πΏ βΆ 7 π₯ β 3 π¦ + 1 7 = 0 2 to the nearest second.

Q13:

Determine the measure of the acute angle between the two straight lines πΏ βΆ β π = ( β 7 , 0 ) + πΎ ( 1 , β 4 ) 1 and πΏ βΆ 1 6 π₯ + 7 π¦ β 2 4 = 0 2 to the nearest second.

Q14:

Let π be the angle between two lines that pass through ( 4 , β 2 ) . If t a n π = 1 2 1 and the slopes of the lines are π and 4 5 π , with π > 0 , find the equations of these lines.

Q15:

Find, to the nearest second, the measure of the acute angle included between the straight line 6 π₯ β 7 π¦ + 4 0 = 0 and the straight line whose slope is 7 3 .

Q16:

Let π be the line on points ( 0 , β 8 ) and ( β 4 , 1 0 ) , and πΏ the perpendicular to π that passes through the origin ( 0 , 0 ) . What is the measure of the positive angle that πΏ makes with the positive π₯ -axis? Give your answer to the nearest second.

Q17:

Determine, to the nearest second, the measure of the positive angle that the straight line πΏ βΆ π₯ = 3 + 1 0 π , π¦ = β 1 5 β 2 1 π makes with the positive π₯ -axis.

Q18:

Determine, to the nearest second, the measure of the positive angle made with the positive π₯ -axis by the perpendicular straight line to the straight line β π = ( 8 , β 9 ) + πΎ ( 1 0 , 7 ) .

Q19:

Find the measure of the acute angle between β ο© ο© ο© ο© β π΄ πΆ and β ο© ο© ο© ο© β π΅ πΆ approximated to the nearest second.

Q20:

Find, to the nearest second, the measure of the angle between the line β 3 π₯ + 4 π¦ β 2 = 0 and the positive π₯ -axis.

Q21:

Find the measure of the acute angle that lies between the straight line whose direction vector is r = β¨ 1 , β 3 β© , and the straight line whose equation is β 2 π₯ β 5 π¦ + 1 = 0 in degrees, minutes, and the nearest second.

Q22:

Find the measure of the acute angle between β ο© ο© ο© ο© β π΄ π΅ and the π₯ -axis approximated to the nearest second.

Q23:

If π is the measure of the acute angle between the two straight lines whose equations are π π₯ β 3 π¦ β 8 = 0 and β π₯ + 3 π¦ + 1 0 = 0 and t a n π = 1 , find all the possible values of π .

Q24:

If points π΄ ( β 6 , 2 ) , π΅ ( β 2 , β 8 ) , and πΆ ( 3 , π¦ ) form a right-angled triangle at π΅ , find the value of π¦ , and then determine the measures of the other two angles to the nearest second.

Q25:

Find the measure of the acute angle between the following pair of straight lines: and .

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