Calculation and Formulas of Angles Between Two Lines
The relationship between two lines can be determined by the angle generated by the two lines. It's the angle formed by the two lines that are being calculated. The acute angle and the obtuse angle are two angles formed by two intersecting lines. We consider the acute angle between two lines while calculating the angle between two lines.
The angle formed by two lines can be used to calculate the angle formed by two sides of a closed polygon. Let's look at the formulas and examples for the angle formed by two lines in a coordinate plane, as well as three-dimensional space.
The topic of straight lines is particularly significant in IIT JEE Mathematics. In competitions such as the IIT JEE, it frequently yields some straight questions. Because the subject is so broad, students should devote enough time to comprehending the numerous concepts. Under straight lines, the angle between two straight lines is a significant head.
When two straight lines intersect, two sets of angles are formed. A pair of acute and another pair of obtuse angles are formed by the intersection. Angles' absolute values are determined by the slopes of crossing lines.
It's also worth noting that if one of the lines intersects with the y-axis, the angle generated by the intersection cannot be determined because the slope of a line parallel to the y-axis is ambiguous.
Angle Between Two Lines Coordinate Geometry
If you know the coordinates of the three points A, B, and C, you can use analytic geometry to calculate the angle between the lines AB and BC.
When the slope of each line is known from the equation, the angle between the two lines may be calculated by calculating the slope of each line and then using it in the formula to get the angle between two lines.
Calculating Angle Between Two Lines in Coordinate Geometry
1. How to Find Angle Between Two Lines
Let us consider three points are given on the x-axis and y-axis whose coordinates are given.
Consider a line whose endpoints have coordinates (x1 y1) and (x2 y2).
The equation of the slope will be
m = y₂ - y₁/x₂ - x₁
m1 and m2 can be calculated by substituting this in the above formula then the values of m1 and m2 can be substituted in the formula given.
tan θ = ± (m₁ - m₂ ) / (1- m₁*m₂)
2. If the Lines are Parallel
(image will be uploaded soon)
The two lines are parallel means, the angle between them is zero Ɵ = 0°
i.e tan Ɵ =0
m₁ - m₂/1+m₁m₂=0
m₁ - m₂=0
m₁ = m₂
The slopes are equal. It shows that the lines are parallel
Note: The value of tan Ɵ will always be positive.
3. If the Lines are Perpendicular
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When the two lines are perpendicular angle between them will be 90° i.e Ɵ=90°
1/tan Ɵ = 0
1+m₁m₂/m₁-m₂ = 0
1+m₁m₂ =0
m₁m₂ = -1
The product of their slope is -1. It shows that the lines are perpendicular.
Formula to Find the Angle Between Two Lines
Consider two nonparallel lines which have slopes m1 and m2 and Ɵ is the angle between the lines, then the formula for finding the angle between the two lines would be:
Derivation of the Formula
Considering the below figure here, we can see that we have a plane with an x-axis and a y-axis.
Two lines L1 and L2 are intersecting each other. Forming one acute and the other obtuse angle. Let us consider the acute angle as Ɵ. (image will be uploaded soon)
Step 1: At first, We have to show that
θ = θ₂ - θ₁
Step 2: For proving above, Let us consider ∆ABC
By angle sum property we can state that
θ + θ₂ + x = 180 ……..(1)
x + θ₂ = 180…………. (2) (as angle x and θ2 forms a linear pair)
Step 3: From equations 1 and 2, we can equate them.
θ + θ₁ + x = x + θ₂ = 180
Subtracting x from both sides
θ + θ₁ + x - x = x + θ₂ - x
we get
θ + θ₁ = θ₂
Step 4: Subtracting θ₁ from both sides
θ + θ₁ - θ₁ = θ₂ - θ₁
We get,
θ = θ₂ - θ₁
Step 5: Applying tangent on both sides
tan θ = tan ( θ₂ - θ₁)
Using the formula of tangents we get
tan θ = tan θ₂ - tan θ₁/1+tan θ₁tanθ₂
From the inclination of a line, we know that tan θ = m
Hence we can substitute tan θ₁ = m₁ and tan θ₂ = m2 we get,
Solved Examples:
Example 1:
If P (2, -1), Q (5, 3), and R (-2, 6) are three points, find the angle between the straight lines PQ and QR.
Solution:
The slope of PQ is given by
m = ( y2 – y1 ) / (x2 – x1)
m = ( 3 – (- 1) ) / (5 – 2 ))
m = 4/3
Therefore, m₁ = 4/3
The slope of QR is given by
m= (6 - 3 ) / (−2−5)
m= 3/-7
Therefore, m₂ = 3/-7
Substituting the values of m2 and m1 in the formula for the angle between two lines we get,
tan θ = ± (m₂ – m₁ ) / (1- m₁ m₂)
tan θ = ± (3/-7) – (4/3) ) / (1- (3/-7)(4/3))
tan θ = ± (37/33)
Therefore, θ = tan -1 (37 / 33)
Example 2:
Find the angle between the following two lines.
Line 1: 4x -3y = 8
Line 2: 2x + 5y = 4
Solution:
Put 4x -3y = 8 into slope-intercept form so you can clearly identify the slope.
4x -3y = 8
3y = 4x - 8
y = 4x / 3 - 8/3
y = (4/3)x - 8/3
Put 2x + 5y = 4 into slope-intercept form so you can clearly identify the slope.
2x + 5y = 4
5y = -2x + 4
y = -2x/5 + 4/5
y = (-2/5)x + 4/5
The slopes are 4/3 and -2/5 or 1.33 and -0.4. It does not matter which one is m₁ or m₂. You will get the same answer.
Let m₁ = 1.33 and m₂ = -0.4
tan θ = ± (m1 – m₂ ) / (1+ m₁*m₂)
tan θ = ± (1.33 - (- 0.4)) / (1- (1.33)*(-0.4))
tan θ = ± (1.73) / (1- 0.532)
tan θ = ± (1.73 ) / (0.468)
tan θ= 3.696
θ = tan⁻¹ (3.69)
Example 3:
Find the acute angle between y = 3x+1 and y = -4x+3
Solution:
m₁= 3 and m₂ = -4
tan θ = ± (m1 – m2 ) / (1+ m₁*m₂)
tan θ = ± (3-(-4) ) / (1+ 3*-4)
tan θ = ± (7 ) / (1+(-12))
tan θ = ± (7 ) / (-11)
tan θ = ± (7/11)
tan θ = 0.636
θ = tan⁻¹ (0.636)
This is all about angles formed between two lines and how to measure them. Focus on the derivation of the formulas to understand the concepts and grab hold of the idea of measuring angles between two lines.
FAQs on The Angle Between Two Lines
1. How to Find Angle Between Two Lines?
Knowing the slope of two lines or the equation of the two lines can be used to compute the angle between them. The acute angle between two lines is the angle formed by the intersection of two lines.
The angle between two lines can be calculated using the trigonometric tangent function and the slope of the two lines.
m1, and m2
respectively. The tangent function formula can be used to calculate the acute angle between the lines.
The angle between the two lines helps to know the relationship between the two lines. This is a measure of the slope between the two lines. When two lines intersect, there are two angles between the lines: acute and obtuse. Here we consider the acute angle between the lines, with respect to the angle between the two lines.
2. What is an angle?
Geometry is a discipline of mathematics that studies shapes and how they are measured. It also focuses on the forms' relative arrangement and spatial features. We know that geometry is divided into two types: 2D and 3D. All geometrical shapes are formed by points, lines, rays, and flat surfaces before they are divided. The measurement between the two lines is called an "Angle" when the two lines or rays converge at a common location.
In plane geometry, an angle is a shape created by two rays or lines that have a common endpoint.
The Latin word "angulus" means "corner," and the term "angle" is derived from it. The sides of an angle are the two rays, and the common terminus is called the vertex. It is not necessary for the angle in the plane to be in Euclidean space. The angle formed by the intersection of two planes in Euclidean or other space is called the dihedral angle.
The symbol "is used to indicate the angle. Greek letters, English letters, etc can be used to represent the angle measurement between the two rays. When angles are measured from a line, we can distinguish between two sorts of angles: positive and negative angles.
3. How to measure the angle?
Angle measurement is an important problem in geometry. A protractor can be used to precisely measure an angle. The term 'degree measure' refers to the measurement of an angle in degrees. Because one full revolution equals 360 degrees, it is divided into 360 parts. A degree is the unit of measurement for each component of the revolution. It will be quite simple to construct an angle using a protractor if we know how to measure the angle.
A protractor can be used to measure any angle, whether acute, obtuse, reflex, or right. Although we can distinguish between acute, obtuse, and reflex angles based on the right angle, we can't distinguish between acute, obtuse, and reflex angles. However, if we had two acute angles to measure, we won't be able to tell which is the bigger. As a result, we must use a protractor to correctly measure the angles.
4. What is Angle Measurement – Degree Measure?
A complete revolution is divided into 360 degrees when the initial and terminal sides are in the same position after spinning clockwise or anticlockwise. The angle is said to have a measure of one degree if the rotation from the initial side to the terminal side is (1360)th of a revolution. It's written as 1°.
Time is measured in hours, minutes, and seconds, with one hour equaling 60 minutes and one minute equaling 60 seconds. In the same way, while measuring angles,
1 degree = 60 minutes denoted as 1° = 60′
1 minute = 60 seconds denoted as 1′ = 60″
5. What are the geometric Characteristics Of Angles?
Acute, right, obtuse, and straight angles are the four types of angles that can be found. An acute angle is more than 0° but less than 90° in degree. An obtuse angle is larger than 90 degrees but less than 180 degrees, while a straight angle, which looks like a straight line, is 180 degrees.
When two angles have the same measurement, they are called congruent angles. They are said to be complementary angles if their sum is 90 degrees. They will be called supplementary angles if their sum is equal to 180°. Rays extending from the vertex can bisect (split in half) or trisect (divide into thirds) angles.
Four angles are formed when two lines intersect. Vertical angles are those that are immediately across from each other and are congruent. Because they share a common side, adjacent angles are referred to as such. The lines are deemed perpendicular or orthogonal if they intersect at a point where each angle measures 90 degrees.
Angles have trigonometric values associated with them, such as sine, cosine, and tangent, in addition to their size. These numbers connect an angle's magnitude to the length of its sides. In fields like navigation, astronomy, and architecture, these values are especially significant.
