Explain two procedures that you can use to draw adjacent angles with given measures.

As you have not  talked about polygon, but let me start from here ,if you are taking a triangle ABC  the  adjacent angles of A are angle B and angle C.

If we are taking Quadrilateral ABCD into consideration the  adjacent angle of A are angle B and angle D.

Now coming to a point , if we have to draw adjacent angles passing through a point then that point is starting point of  three rays .

Step-by-step explanation:

Adjacent angles :Two angles are said to be adjacent if their non common arms lie on opposite side of common arm. In terms of rays the two rays should lie on opposite side of third ray.Suppose OA , OB and OC are three rays, taking OB as third ray OA and OC will lie on opposite side of ray OB.

1. When two angles are supplementary

 First draw a line. Mark a point P on it. Mark two points S and T on opposite sides of Point P. From P draw a ray P Q.The two angles ∠QPS and ∠QPT are adjacent angles.

2. Suppose two angles are ∠AOB=60° and ∠AOC=60°. Ray OA is common.Ray OB and Ray OC will lie in opposite sides of Ray OA.

Or

Draw an angle of 120°, draw it's bisector .it's bisector will be Ray OA.

I have depicted pictures for this.  

Also if ∠AOB=30° and ∠AOC=45°,taking OA as a common ray , Ray OB  and ray OC will lie on opposite side of common ray OA.

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