# Theorems on Linear Pair and Vertically Opposite Angles

**Linear Pair Axiom :-**

**Axiom1**

If a ray stands on a line , then the sum of two adjacent angles so formed is 180**Â°**

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**Axiom2**

If the sum of two adjacent angles is 180**Â° ** then the non common arms of the angles form a line.

Consider the angles formed by rays OA, OB, OC and OD, having a common end point O. Then the four angles together form a complete angle, i.e.

âˆ 1 + âˆ 2 + âˆ 3 + âˆ 4 = 360Â°

In general, the sum of all the angles at a point is 360Â° or 4 right angles.

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**Example : **Find the angles x, y and z in the figure.

âˆ x + 130Â° = 180Â° (Linear pair)

Therefore âˆ x = 180Â° - 130Â° = 50Â°

âˆ y = 50Â° (since x and y are vertically opposite angles, âˆ x = âˆ y)

âˆ z =180Â° - âˆ y (Linear pair)

= 180Â°-50Â°= 130Â°

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**Example : **Find the angles in the figure.

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x + 2x - 30Â° = 90Â° (Complementary angles)

3x = 120Â° or x =40Â°

Therefore 2x - 30Â° = 80Â° - 30Â° = 50Â°

The angles are 40Â° and 50Â°.

**Theorem 1**

If two lines intersect each other, then vertically opposite angles are equal.

**Given: ** Two lines AB and CD intersect each other at O.

To Prove: âˆ AOD = âˆ COB and âˆ AOC = âˆ DOB

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**Proof: ** Ray OD stands on the line AB

âˆ´ âˆ AOD + âˆ DOB = 180Â° â€¦.(i)

Since ray OA stands on the line CD

âˆ´ âˆ COA + âˆ AOD = 180Â° â€¦. (ii)

Comparing equation (i) and (ii), we get

âˆ AOD + âˆ DOB = âˆ COA + âˆ AOD

âˆ´ âˆ DOB = âˆ COA

Similarly we can prove that

âˆ AOD = âˆ COB

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In the figure, lines *l*_{1} and *l*_{2} intersect at O, forming angles as shown. If x = 53Â°, find the values of *y, z* and *p.*

Since *l*_{1} and *l*_{2} intersect each other

Â ... âˆ x = âˆ z (Vertically opposite angles)

... 53Â° = âˆ z

...Â z = 53Â°

Since y and z form a linear pair

... âˆ y + âˆ z = 180Â°

...âˆ y + 53Â° = 180Â°

...âˆ y = 180Â° - 53Â° = 127Â°

Again, âˆ y = âˆ p (Vertically opposite angles)

...Â 127Â° = âˆ p or p = 127Â°

Hence x = z = 53Â° and y = p = 127^{o}

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