<\/span><\/p>\n If the angle between two intersecting lines having slopes m_1 \\text{ and } m_2<\/span><\/span>\u00a0is \u03b8, then the formula for the angle \u03b8 is given by \\tan \\theta=\\pm\\left(m_{2}-m_{1}\\right) \/\\left(1+m_{1} m_{2}\\right)<\/span><\/span><\/p>\n <\/p>\n Consider the diagram shown below:<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n In this diagram, two lines intersect at a point.<\/span><\/p>\n Let the slope of these lines be m_{1} \\text { and } m_{2}<\/span> so\u00a0<\/span><\/p>\n \\tan \\theta_{1}=m_{1} \\text { and } \\tan \\theta_{2}=m_{2}<\/span><\/span><\/p>\n We know that the sum of all the angles in a triangle is equal to 180 degrees.<\/span><\/p>\n So, in \u25b3ABC,\u00a0<\/span><\/p>\n \\theta +{\\theta}_1+\\text{x}<\/span><\/span>\u00a0= 180\u00b0 (Equation No. 1)<\/span><\/p>\n Also, \\text{x} +{\\theta}_2<\/span> <\/span>= 180\u00b0 (since \\text{x and } {\\theta}_2<\/span><\/span>\u00a0produce a straight line)<\/span><\/p>\n Substituting this value in equation no. 1<\/span><\/p>\n \\theta +{\\theta}_1+\\text{x}=\\text{x}+{\\theta}_2<\/span><\/span><\/p>\n \u21d2 \\theta +{\\theta}_1={\\theta}_2<\/span><\/span><\/p>\n Or, \\theta=\\theta_{2}-\\theta_{1}<\/span><\/p>\n Therefore, \\tan \\theta = \\tan (\\theta_2 -\\theta 1)<\/span><\/span><\/p>\n \u00a0\u21d2 \\tan \\theta = (\\tan \\theta_2 -\\tan \\theta_1)\/(1+\\tan\\theta_1\\tan\\theta_2)<\/span><\/span><\/p>\n Now, substitute the values of \\tan \\theta_{1} \\text { and } \\tan \\theta_{2}\\text{ as }m_1\\text{ and }m_2<\/span><\/span>,<\/span> respectively,\u00a0<\/span><\/p>\n We have<\/span> \\tan \\theta = (m_2 -m_1)\/(1+m_1m_2)<\/span><\/span><\/b><\/p>\n Note that in this equation, the value of tan \u03b8 will be positive if \u03b8 is acute and negative if \u03b8 is obtuse.<\/span><\/p>\n <\/span><\/p>\n If the two lines are perpendicular, the angle between them will be 90 degrees. In this case,<\/span><\/p>\n \u00a0 \u00a0 \u00a0 1\/\\tan\\theta=0<\/span><\/span><\/p>\n \u00a0\u21d2 <\/span>(1+m_{1} m_{2})\/(m_{2}-m_{1})=0<\/span><\/p>\n \u00a0\u21d2 <\/span>(1+m_{1} m_{2})=0 <\/span> If the product of the slopes of lines is -1, it represents that the lines are perpendicular.<\/p>\n <\/p>\n If two lines are parallel, the angle between them will be zero degrees. In this case,<\/span><\/p>\n \u00a0 \u00a0 \u00a0 \\tan\\theta=0<\/span><\/span><\/p>\n \u00a0\u21d2 \\left(m_{2}-m_{1}\\right) \/ 1+m_{1} m_{2}=0 <\/span> If the slopes of the two lines are equal, it shows that the lines are parallel.\u00a0\u00a0<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n The slope of XY is given by<\/span><\/p>\n m_{1}=\\left(y_{2}-y_{1}\\right) \/\\left(x_{2}-x_{1}\\right)<\/span> Substituting the values, we get<\/span><\/p>\n \\mathrm{m}_{1}=\\{3-(-1)\\} \/(5-2) <\/span> Similarly, the slope of YZ is given by<\/span><\/p>\n m_{2}=(6-3) \/(-2-5) <\/span> We know that the formula for the angle between two lines<\/span><\/p>\n \\tan \\theta=\\left(m_{2}-m_{1}\\right) \/\\left(1-m_{1} m_{2}\\right)<\/span><\/span><\/p>\n We will substitute the values of \\mathrm{m}_{1} \\text { and } \\mathrm{m}_{2}<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 tan \u03b8 = [{3\/(-7) \u2013 (4\/3)} \/ {(1+ (-3\/7)(4\/3)}]<\/span><\/p>\n \u00a0\u21d2 tan \u03b8 = [(-37\/21) \/ (-3\/7)]<\/span><\/p>\n \u00a0\u21d2 tan \u03b8 = (37\/9)<\/span><\/p>\n Therefore,\u00a0 \\theta=\\tan^{-1}(37\/9)<\/span><\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n We will have to convert the equations of both the lines into a slope-intercept form (y = mx + c) so that we can identify the slope.<\/span><\/p>\n Starting with the first equation<\/span><\/p>\n \u00a0 \u00a0 \u00a04x – 3y = 8<\/span><\/p>\n \u00a0\u21d2 3y = 4x – 8<\/span><\/p>\n \u00a0\u21d2 y = (4x – 8)\/3<\/span><\/p>\n \u00a0\u21d2 y = (4x\/3) – (8\/3)<\/span><\/p>\n \u00a0\u21d2 y = (4\/3)x – (8\/3)<\/span><\/p>\n Similarly we will convert 2x + 5y = 4 into slope-intercept form.<\/span><\/p>\n \u00a0 \u00a0 \u00a02x + 5y = 4<\/span><\/p>\n \u00a0\u21d2 5y = -2x + 4<\/span><\/p>\n \u00a0\u21d2 y = (-2x +4)\/5<\/span><\/p>\n \u00a0\u21d2 y = (-2x\/5) + (4\/5)<\/span><\/p>\n \u00a0\u21d2 y = (-2\/5)x + (4\/5)<\/span><\/p>\n Now comparing both the equations with mx + c, we get the values of slopes.<\/span><\/p>\n m_1<\/span><\/span>= 4\/3 = 1.33 and m_2<\/span> <\/span>= (-2\/5) = -0.4<\/span><\/span><\/p>\n We know that the formula for the angle between two lines<\/span><\/span><\/p>\n \\tan \\theta=\\left(\\mathrm{m}_{2}-\\mathrm{m}_{1}\\right) \/\\left(1+\\mathrm{m}_{1} \\mathrm{~m}_{2}\\right) <\/span> <\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\nAngle Between Two Lines: Formula<\/b><\/h2>\n
Angle Between Two Straight Lines Derivation<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Angle-Between-Two-Lines-01-300x249.png\")
<\/span><\/p>\nIf the Lines are Perpendicular<\/b><\/h3>\n
\u00a0\u21d2 <\/span>m_{1} m_{2}=-1<\/span><\/p>\nIf the Lines are Parallel<\/b><\/h3>\n
\u21d2 \\left(m_{2}-m_{1}\\right)=0 <\/span>
\u21d2 m_{1}=m_{2}<\/span><\/span><\/p>\nExamples\u00a0<\/b><\/h2>\n
1. If X (2, -1), Y (5, 3), and Z(-2, 6) are three points, find the angle between the straight lines XY and YZ?<\/span><\/h3>\n
Here, x_{1}=2, y_{1}=-1, x_{2}=5, y_{2}=3<\/span><\/span><\/p>\n
\\mathrm{m}_{1}=(4 \/ 3) <\/span>
\\text { Hence, } \\mathrm{m}_{1}=4 \/ 3<\/span><\/span><\/p>\n
m_{2}=3 \/-7 <\/span>
\\text { Hence, } m_{2}=-(3 \/ 7)<\/span>
<\/span><\/span><\/p>\n2. Find the angle between the two lines 4x – 3y = 8 and 2x + 5y = 4.<\/span><\/h3>\n
\u00a0\u21d2 \\tan \\theta=\\{(1.33-(-0.4)\\} \/\\{1+(1.33) \\times(-0.4)\\} <\/span>
\u00a0\u21d2 \\tan \\theta=(1.73) \/(1-0.532) <\/span>
\u00a0\u21d2 \\tan \\theta=(1.73) \/(0.468) <\/span>
\u00a0\u21d2 \\tan \\theta=3.696<\/span>
\\therefore \\theta=\\tan ^{-1}(3.696)<\/span><\/span><\/p>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n