Notes
Parallel lines : The lines which are coplanar and do not intersect each other are called parallel lines.
Transversal and intercept :
Transversal : A line intersecting two other coplanar lines in two distinct points is called a transversal of the two lines.
- In the figure, line a and line b are the transversal of lines x and y.
- Line x and line y are the transversals of lines a and b.
Intercept : A segment cut off by a transversal on two distinct lines is called an intercept.
In the above figure, seg PQ is the intercept made by the transversal a on the lines x and y.
Seg PR is the intercept made by the transversal y on the lines a and b.
Angles formed by the transversal :
In the figure, line n is the transversal of lines l and m.
∠a, ∠b, ∠c and ∠d are formed at the point A.
∠e, ∠f, ∠g and ∠h are formed at the point B.
Thus, eight angles are formed. Of them, one side or ray of each angle is contained in the transversal.
Here, in all 8 angles are formed. Pairs of angles formed out of these angles are as follows :
Pairs of corresponding angles :
(i) ∠d, ∠h (ii) ∠a, ∠e (iii) ∠c, ∠g (iv) ∠b, ∠f
Pairs of alternate interior angles :
(i) ∠c, ∠e (ii) ∠b, ∠h
Pairs of alternate exterior angles :
(i) ∠d, ∠f (ii) ∠a, ∠g
Pairs of interior angles on the same side of the transversal :
(i) ∠c, ∠h (ii) ∠b, ∠e
Some important properties :
- When two lines intersect, the pairs of opposite angles formed are congruent.
- The angles in a linear pair are supplementary.
- When one pair of corresponding angles is congruent, then all the remaining pairs of corresponding angles are congruent.
- When one pair of alternate angles is congruent, then all the remaining pairs of alternate angles are congruent.
- When one pair of interior angles on one side of the transversal is supplementary, then the other pair of interior angles is also supplementary.
Interior angle theorem :
Theorem : If two parallel lines are intersected by a transversal, the interior angles on either side of the transversal are supplementary.
Given : line l || line m and line n is their transversal.
Hence as shown in the figure
∠a, ∠b are interior angles formed on one side and
∠c, ∠d are interior angles formed on other side of the transversal.
To prove : ∠a + ∠b = 180°
∠d + ∠c = 180°
Corresponding angles and alternate angles theorems :
Theorem : The corresponding angles formed by a transversal of two parallel lines are of equal measure.
Given : line l || line m line n is a transversal.
To prove : ∠a = ∠b
Theorem : The alternate angles formed by a transversal of two parallel lines are of equal measures.
Given : line l || line m line n is a transversal.
To prove : ∠d = ∠b
Use of properties of parallel lines :
Let us prove a property of a triangle using the properties of angles made by a transversal of parallel lines.
Theorem : The sum of measures of all angles of a triangle is 180°.
Given : Δ ABC is any triangle.
To prove : ∠ABC + ∠ACB + ∠BAC = 180°.
Construction : Draw a line parallel to seg BC and passing through A. On the line take points P and Q such that, P - A - Q.
Tests for parallel lines :
Whether given two lines are parallel or not can be decided by examining the angles formed by a transversal of the lines.
- If the interior angles on the same side of a transversal are supplementary then the lines are parallel.
- If one of the pairs of alternate angles is congruent then the lines are parallel.
- If one of the pairs of corresponding angles is congruent then the lines are parallel.
Interior angles test :
Theorem : If the interior angles formed by a transversal of two distinct lines are supplementary, then the two lines are parallel.
Given : Line XY is a transversal of line AB and line CD.
∠BPQ + ∠PQD = 180°
To prove : line AB || line CD
Alternate angles test :
Theorem : If a pair of alternate angles formed by a transversal of two lines is congruent then the two lines are parallel.
Given : Line n is a transversal of line l and line m.
∠a and ∠b is a congruent pair of alternate angles.
That is, ∠a = ∠b
To prove : line l || line m
Corresponding angles Test :
Theorem : If a pair of corresponding angles formed by a transversal of two lines is congruent then the two lines are parallel.
Given : Line n is a transversal of line l and line m.
∠a and ∠b is a congruent pair of corresponding angles.
That is, ∠a = ∠b
To prove : line l || line m
Corollary I : If a line is perpendicular to two lines in a plane, then the two lines are parallel to each other.
Given : Line n ⊥ line l and line n ⊥ line m
To prove : line l || line m
Corollary II : If two lines in a plane are parallel to a third line in the plane then those two lines are parallel to each other. Write the proof of the corollary.
Given : Lines AB, CD and EF are coplanar such that line AB || line CD and line AB || line EF. Line PT is the transversal intersecting lines AB, CD, EF at points Q, R and S respectively.
To prove : line CD || line EF.