AXIOM1 : If a ray stands on a line, then the sum of two adjacent angles so formed is 180�.
This gives us another definition of linear pair of angles – when the sum of two adjacent angles is 180�, then they are called as linear pair of angles. In figure, a ray PQ standing on a line ∠forms a pair of adjacent angles ∠ BPQ and ∠ QPA.
The sum of the measure of the two angles is a straight angle which is equal to 180�.
The ray PB and PA are opposite rays. Thus, the two angles form a linear pair of angles. The above axiom can be stated in the reverse way as below:
AXIOM2 : If the sum of two adjacent angles is 180�, then the noncommon arms of the angles form a line. In figure, angles ∠BAC and ∠CAD are adjacent angles such that ∠BAC + ∠CAD = 180�
Then, we have
∠BAD = 180� (∵ ∠BAD = ∠BAC + ∠CAD)
Thus, ∠BAD is a straight angle. Therefore, AB and AD are in one line. The axiom 1 and 2 are called linear pair axioms. The two axioms are reverse of each other.
(x) Vertically Opposite Angles : If two lines intersect each other then, the pairs of opposite angles formed are called vertically opposite angles. Two lines AB and CD intersect at point O.
Then, two pairs of vertically opposite angles are formed.
One pair is ∠AOD & ∠BOC. The other pair is ∠AOC & ∠BOD.
THEOREM1 : If two lines intersect each other, then the vertically opposite angles are equal.
Given : Two lines AB and CD intersect at a point O.
Two pairs of vertically opposite angles are :
(i) ∠AOC and ∠BOD
(ii) ∠AOD and ∠BOC
To Prove : (i) ∠AOC =∠BOD
(ii) ∠AOD = ∠BOC i.e. two pairs of vertically opposite equal angles.
Proof :
Ex. In figure, ∠AOC = 50^{o} and ∠BOC = 20^{o} , find ∠AOB
Sol.
Ex. In figure, are complementary angles. Find the value of x and hence find
Sol. In figure, We have, ∠AOC + ∠BOC = 90�
⇒ (x+ 10)� + (2x + 5)� = 90�
⇒ 3x + 15� = 90�
⇒ 3x = 90� – 15� = 75�
⇒ x = 25�
Now, ∠AOC = (25 + 10)� = 35� and ∠BOC = {2 � 25 + 5}� = 55�
Ex. In figure, ray AD stands on the line CB, ∠BAD = (2x + 10)� and ∠CAD = (5x + 30)�, find the value of x and also write ∠BAD and ∠CAD.
Sol. ∠BAD + ∠CAD = 180� (Linear pair of angles)
⇒ (2x + 10)� + (5x + 30)� = 180�
⇒ 7x + 40� = 180�
⇒ 7x = 180� – 40�
⇒ 7x = 140�
⇒ x = 140/7
⇒ x = 20
Now, ∠BAD = (2 � 20 + 10)� = 50�
and ∠CAD = (5 � 20 + 30)� = 130�
Ex. In figure, ray OC stands on the line AB and ∠BOC = 125�. Find reflex ∠AOC.
Sol. In figure, ∠AOC and ∠BOC are linear pair angles.
⇒ ∠AOC + ∠BOC = 180�
⇒ ∠AOC + 125� = 180�
⇒ ∠AOC = 180� – 125� = 55� (∵ ∠BOC = 125�)
Now, Reflex ∠AOC = 360� – ∠AOC = 360� – 55� = 305�
CORRESPONDING ANGLES AXIOM: If a transversal intersects two parallel lines, then each pair of corresponding angles are equal.
Conversely, if a transversal intersects two lines, making a pair of equal corresponding angles, then the lines are parallel.
By using the above axiom, we can now deduce the other facts about parallel lines and their transversal.
THEOREM2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
Given : AB and CD are two parallel lines and a transversal EF intersects them at point G and H respectively. Thus, the alternate 1
interior angles are ∠2 and∠1, and ∠3 and ∠4.
To Prove : ∠1 = ∠2 and ∠3 = ∠4
Proof :
STATEMENT  REASON  
1. 2. 3. 4. 5. 6.  ∠2 =∠6 ∠1 = ∠6 ∠2 = ∠6 and ∠1 = ∠6 ⇒1 = ∠2 Similarly ∠4 = ∠5 ∠3 = ∠5 ∠4 = ∠5 and ∠3 = ∠5 ⇒3 = ∠4  Vertically opposite angles By corresponding angles axiom From equation (1) and (2) Vertically opposite angles By corresponding angles axiom From equation (4) and (5) 
Hence, proved.
THEOREM3 (converse of theorem of 2) : If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel.
Given : A transversal 't' intersects two lines AB and CD at P and Q respectively such that ∠1 and ∠2 are a pair of alternate interior angles and ∠1 = ∠2.
To Prove : AB║CD
Proof :
STATEMENT  REASON  
1  ∠2 = ∠3  Vertically opposite angles 
2  ∠1 = ∠2  Given 
3  ∠2 = ∠3 and ∠1 = ∠2 ⇒ ∠1 = ∠3  From (1) and (2) 
4  AB║CD  By converse of corresponding angles axiom 
Hence, proved.
THEOREM4 : If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
Given : AB and CD are two parallel lines. Transversal "t" intersects AB at P and CD at Q, making two pairs of
consecutive interior angles, ∠1, ∠2 and ∠3, ∠4.
To Prove :
(i) ∠1 + ∠2 = 180� and
(ii) ∠3 + ∠4 = 180�
Proof :
STATEMENT  REASON  
1.  ∵ AB ║ CD ∴ ∠1 = ∠5  Corresponding ∠s axiom 
2.  ∠5 + ∠2 = 180�  Linear pair of angles 
3.  ∠1 + ∠2 = 180�  From (1) and (2) 
4.  ∵ AB ║ CD ∴ ∠3 = ∠6  Corresponding ∠s axiom 
5.  ∠6 + ∠4 = 180�  Linear pair of angles 
6.  ∠3 + ∠4 = 180�  From (4) and (5) 
Hence, proved.
THEOREM5 (converse of theorem 4) : If a transversal intersects two lines in such a way that a pair of consecutive interior angles are supplementary, then the two lines are parallel.
Given : A transversal t intersect two lines AB and CD at P and Q respectively such that ∠1 and ∠2 are a pair of
consecutive interior angles, and ∠1 + ∠2 = 180�.
To Prove : AB║CD
Proof :
STATEMENT  REASON  
1.  ∠1 + ∠2 = 180�  Given 
2.  ∠2 + ∠3 = 180�  Linear pair of angles 
3.  ∠1 + ∠2 = ∠2 + ∠3 ⇒ ∠1 = ∠3  From (1) and (2) 
4.  ⇒ AB ║ CD  By converse of corresponding angles axiom 
Hence, proved.
THEOREM6 : If two lines are parallel to the same line, they will be parallel to each other.
Given : Line m║ line and line n ║ line .
To Prove : Line m║ line n.
Construction : Draw a line t transversal for the lines , m and n.
Proof :
STATEMENT  REASON  
1.  ∵ m║ ∴ ∠1 = ∠2  Corresponding angles 
2.  ∵ n║ ∴ ∠1 = ∠3  Corresponding angles 
3.  ∠1 =∠2 and ∠1=∠3 ⇒ ∠2=∠3  From (1) and (2) 
4.  ⇒ m║n  By converse of corresponding angles axiom 
Hence, proved.
ANGLE SUM PROPERTY OF A TRIANGLE
In previous classes, we have learnt that the sum of the three angles of a triangle is 180�. In this section, we shall prove this fact as a theorem.
THEOREM7 : The sum of all the angles of a triangle is 180�.
Given : In a Δ PQR, P
∠1, ∠2 and ∠3 are the angles of Δ PQR.
To Prove : ∠1 + ∠2 + ∠3 = 180�
Construction :
Draw a line XPY parallel to QR through the opposite vertex P.
Proof :
STATEMENT  REASON  
1.  ∠4 + ∠1 + ∠5 = 180�  XPY is a line 
2.  ∵ XPY ⊥ QR ∴ ∠4 = ∠2 and∠5 = ∠3  Alternate interior angles 
3.  ∠2 + ∠1 +∠3 = 180� or ∠1 + ∠2 + ∠3 = 180�  From (1) and (2) 
Hence, proved.
EXTERIOR ANGLE OF A TRIANGLE
Consider the ΔPQR. If the side QR is produced to point S, then ∠PRS is called
an exterior angle of ΔPQR.
The ∠PQR and ∠QPR are called the two interior opposite angles of the exterior ∠PRS.
THEOREM8 : If a side of a triangle is produced, then the exterior P
angle so formed is equal to the sum of the two interior opposite angles.
Given : In a Δ PQR, ∠1, ∠2 and ∠3 are the angles of ΔPQR, side QR is produced to S and exterior angle ∠PRS = ∠4.
To Prove : ∠4 = ∠1 + ∠2
Proof :
STATEMENT  REASON  
1.  ∠3 + ∠4 = 180�  Linear pair of angles 
2.  ∠1 + ∠2 + ∠3 = 180�  The sum of all the angles of a triangles is 180� 
3.  ∠3 + ∠4 = ∠1 + ∠2 + ∠3 ⇒∠4 = ∠1 + ∠2.  From (1) and (2) 
Hence, proved.
1 videos228 docs21 tests

1. What is the Angle Sum Property? 
2. How can I prove the Angle Sum Property? 
3. What are axioms in the context of Lines and Angles? 
4. Can the Angle Sum Property be extended to other polygons? 
5. How is the Angle Sum Property useful in solving geometry problems? 
1 videos228 docs21 tests


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