It is given that OA and OX are opposite rays. C,D,and E are points outside of line AX.and on the same side of it . Do the following pairs of angles 

form a linear pair or not?
(1) angle COA and angle COX (ii) angle COA and angle COD (iii) angle COX and angle COE .
Explain the answers with proper illustrations.What is the key point that makes the decide if a pair of angles form a linear pair or not ?

Expert Answers

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linear pair: A linear pair of angles is formed when two lines intersect. The two lines form a pair of angles that must sum to 180 degrees.

opposite rays: Two rays with a common endpoint that point in opposite directions and form a straight line (i.e. 180 degrees).

A <--------------O----------->X      OA and OX are opposite rays.


First, we must draw the picture. Either above or below the line AX, draw and label three points: C, D, and E. Now draw a line from C to O, then from D to O, etc.

Now we can clearly see the angles COA, COX, COE, etc., and we can answer the questions.

i) Do angles COA and COX form a linear pair? Yes, because two lines come together (AX and CO) to form two angles that sum to 180 degrees.

ii) Angles COA and COD? No, because the angle COA is less than 180 degrees, therefore the angles created by DO can not sum to 180.

iii) Angles COX and COE? What do you think? Is the angle XOE equal to 180 degrees?

The key point that determines whether three rays that share a common endpoint form a linear pair is: are any two of those rays "opposite rays".

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