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.