Lines and Angles - Kamal Lohia
Faculty and Content Developer at Tathagat | Delhi College of Engineering
A line, mathematically, is a straight line with no end points. So it has infinite length and zero thickness.
A ray, on the other hand, is a line with one end point. And it is used to form angles. Its length is also infinite.
A line segment is a part of line with both end points. Certainly its length is finite. And in common usage, we address a line segment as line. But CAT may trap you on it. So beware. :)
How do you determine whether two lines are parallel or not?
You may answer: if they do not intersect.
Or someone may also say that if distance between them is constant.
Let’s frame it in a beautiful data sufficiency question.
Are the lines P and Q are parallel?
I. P and Q do not intersect.
II. Perpendicular distance of any point on P from Q is constant.
If you are thinking that question can be answered by using either statement alone, them I am sorry but you are wrong. Two lines in different parallel planes will certainly never intersect. But it is not compulsory that distance between them is constant. Now important point is that, if two lines are parallel then there is certainly a plane which passes through both of them.
Another type of problems associated with lines
How many regions/parts are created when a line is drawn in a plane?
Certainly a line will divide the plane in two parts.
Next…In how many regions/parts a plane is divided by two lines?
Now it depends on whether the two lines intersect or not. If they don’t intersect, number of parts created is 3 and if the lines intersect then number of parts is 4.
But what if question asks to find the number of parts created by n lines in a plane? :)
Clearly, as we saw earlier, number of parts (P) created depends on number of lines (L) and number of intersection (I) of two lines. Now the relation between them (without proof) is:
P = L + I + 1
Important point to keep in mind is that I is the intersection of two lines at a point. If more than two lines pass through same point then we need to increase I by number of extra lines passing through same point.
10 chords are drawn in a circle. Find the maximum number of parts in which circle is divided.
For maximum parts, every two chords must intersect and intersect at different point.
So answer becomes, P = L + I + 1 = 10 + C (10, 2) + 1 = 56
According to measurement, angles can be classified as:
There are two more classification –
Supplementary Angles – two angles whose sum is 1800 are said to be supplement of each other.
Complementary Angles – two angles whose sum is 900 are said to be complement of each other.
Now come angles formed by intersection of lines and parallels intersected by transversals.
In the above figure, l and m are two parallel lines intersected by a transversal PS. The following properties of the angles can be observed:
< 1 = < 7 and < 2 = < 8 [AlternateExterior Angles]
< 3 = < 5 and < 4 = < 6 [Alternate Interior angles]
< 1 = < 5, < 2 = < 6, < 4 = < 8, < 3 = < 7 [Corresponding angles]
< 4+ < 5= < 3+ < 6=180o [Co-interior angles are interior angles on same side of the transversal that adds up to 180o]
< 1+ < 8= < 2+ < 7=180o
< 1+ < 8= < 2+ < 7=180o [Co-exterior angles are exterior angles on same side of the transversal that adds up to 180o]
Besides these, we discriminate triangles according to their angles.
Acute angled Triangle – have all three angles acute.
Right angled Triangle – have one right angle and other two acute.
Obtuse angled Triangle – have one obtuse angle and other two acute.
Also, if in a triangle two angles are equal, then sides opposite to those equal angles are also equal in length. And, if in a triangle two angles are unequal, then side opposite to smaller angle is smaller.
We can also find a direct relationship between side lengths and angles of a triangle – given by
Sine rule as below:
Another important one is Cosine rule:
Where a, b, c are the side lengths opposite to vertices A, B and C respectively as shown below. And R is the radius of circumscribe circle of the triangle ABC.
Two basic properties related to angles in triangle:
Angle Sum Property – Sum of all the three angles of a triangle is 180o.
Exterior Angle Property – Exterior angle at a vertex of a triangle is equal to sum of two opposite angles.