In this chapter we will prove that sum of all adjacent angles in a straight lines measures 180 degree.
AB is a straight line intersected by ray OM in between.
The three angles formed in the above diagram are ∠AOM, ∠BOM & ∠AOB.
We know that angle of straight line is 180 degree.
∠AOB = 180 degree.
From the above figure we can write as;
∠AOM + ∠BOM = ∠AOB
∠AOM + ∠BOM = 180 degree
Hence we proved that in straight line, sum of adjacent angle is 180 degrees.
Using the above proof, we have two axioms in geometry.
Axiom 01
If a ray stands on a straight line then the sum of two adjacent angle formed will be 180 degree.
Axiom 02
If the sum of two adjacent angles is 180 degree then the non common arm of angle makes a straight line.
I hope you understood the above concept. Let us solve some problems related to it.
Question 01
In below figure, AB is a straight line and ray OM intersect the line at point O. Line OR & OT are angle bisector of ∠AOM and ∠BOM respectively. Find the measure of ∠ROT.
Solution
It’s given that line OR is angle bisector of ∠AOM.
∠ROM = ∠ AOR
∠ ROM = x
Similarly line OT is angle bisector of ∠BOM.
∠MOT = ∠BOT
∠ MOT = y
We know that in straight line, the sum of all adjacent angle is 180 degree.
∠ROM + ∠AOR + ∠MOT + ∠BOT = 180 degree
x + x + y + y = 180
2x + 2y = 180
x + y = 90 degree
Now let’s find the value of ∠ROT.
∠ ROT = ∠x + ∠y
∠ ROT = 90 degree
Hence, the measure of ∠ROT is 90 degrees.
Question 02
In the below figure, OP, OQ, OR & OS are rays meeting at point O.
Prove that ∠POQ + ∠ROQ + ∠SOR + ∠SOP = 360 degree
Solution
To solve the above question, extend line OQ to point M as shown in below figure.
Since MQ is a straight line, the angle adjacent to one side of the ray measures 180 degree.
∠MOP + ∠QOP = 180 degree
Similarly;
∠MOS + ∠SOR + ∠ROQ = 180 degree
Adding both the equations we get;
∠MOP + ∠QOP + ∠MOS + ∠SOR + ∠ROQ = 180 + 180 degree
∠MOP + ∠QOP + ∠MOS + ∠SOR + ∠ROQ = 360 degree
From the above figure we can say that;
∠ SOM + ∠ MOP = ∠ SOP
Putting this expression in above equation, we get;
∠SOP + ∠QOP + ∠SOR + ∠ROQ = 360 degree
Hence Proved.