Chapter 6.Lines and Angles
- Basic Terms and Definitions
I.a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by AB
, and a line is denoted by AB
II. If three or more points lie on the same line, they are called collinear points; otherwise they are called non-collinear points.
III an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle.
(i)acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 1 Types of Angles
Note:-
1.An acute angle measures between 0° and 90°,
2. whereas a right angle is exactly equal to 90°.
3.An angle greater than 90° but less than 180° is called an obtuse angle.
4.Also, recall that a straight angle is equal to 180°.
5.An angle which is greater than 180°but less than 360° is called a reflex angle.
6.Further, two angles whose sum is 90° arecalled complementary angles,
7. and two angles whose sum is 180° are called supplementary angles.
Axiom 1 : If a ray stands on a line, then the sum of two adjacent angles so
formed is 180°.
Axiom 2 : If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
Theorem 1 : If two lines intersect each other, then the vertically opposite
angles are equal.
Proof : In the statement above, it is given that ‘two lines intersect each other’. So,let AB and CD be two lines intersecting at O as in Fig. 6.8. They lead to two pairs of
vertically opposite angles, namely,
(i) Ð AOC and Ð BOD (ii) Ð AOD and O
Ð BOC.
We need to prove that Ð AOC = Ð BOD C
and Ð AOD = Ð BOC.
Now, ray OA stands on line CD. B
Therefore, Ð AOC + Ð AOD = 180° (Linear pair axiom) (1)
Can we write Ð AOD + Ð BOD = 180°? Yes! (Why?) (2)
From (1) and (2), we can write
Ð AOC + Ð AOD = Ð AOD + Ð BOD
This implies that Ð AOC = Ð BOD (Refer Section 5.2, Axiom 3)
Similarly, it can be proved that ÐAOD = ÐBOC
Axiom 3 : If a transversal intersects two parallel lines, then each pair of
corresponding angles is equal.
Axiom 4 : If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other
Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of
alternate interior angles is equal.
Now, using the converse of the corresponding angles axiom, can we show the two lines parallel if a pair of alternate interior angles is equal? the transversal PS intersects lines AB and CD at points Q and R respectively such that
Ð BQR = Ð QRC.
Is AB || CD?
Ð BQR = Ð PQA (Why?) (1)
But, Ð BQR = Ð QRC (Given) (2)
So, from (1) and (2), you may conclude that
< PQA = < QRC
But they are corresponding angles.
So, AB || CD (Converse of corresponding angles axiom)
This result can be stated as a theorem given below:
Theorem 6.3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
In a similar way, you can obtain the following two theorems related to interior angles on the same side of the transversal.
Theorem 6.4 : If a transversal intersects two parallel lines, then each pair of
interior angles on the same side of the transversal is supplementary.
Theorem 6.5 : If a transversal intersects two lines such that a pair of interior
angles on the same side of the transversal is supplementary, then the two lines are parallel.
Theorem 6.6 : Lines which are parallel to the same line are parallel to each
other.
7 Angle Sum Property of a Triangle
the sum of all the angles of a triangle is 180°
Theorem 6.7 : The sum of the angles of a triangle is 180º.
Theorem 6.8 : If a side of a triangle is produced, then the exterior angle so
formed is equal to the sum of the two interior opposite angles.
It is obvious from the above theorem that an exterior angle of a triangle is greater than either of its interior apposite angles.
Summary
1. If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and viceversa.
This property is called as the Linear pair axiom.
2. If two lines intersect each other, then the vertically opposite angles are equal.
3. If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal,
(ii) each pair of alternate interior angles is equal,
(iii) each pair of interior angles on the same side of the transversal is supplementary.
4. If a transversal intersects two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal, or
(iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.
5. Lines which are parallel to a given line are parallel to each other.
6. The sum of the three angles of a triangle is 180°.
7. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.