If AB is parallel RS R_,__,__,__....,. sand O is the mid-point yof AS, then show that ,(i) !!,.AOB = !!,.SOR A B(ii) 0 is mid-point of BR.

To prove that (i) ∠AOB = ∠SOR and (ii) O is the midpoint of BR, we can use the properties of parallel lines and the midpoint theorem.

(i) ∠AOB = ∠SOR:

1. Given that AB is parallel to RS, we can conclude that ∠ABO and ∠SOR are corresponding angles.
2. Since AB is parallel to RS, ∠ABO and ∠SOR are also alternate interior angles.
3. By the alternate interior angles theorem, we know that corresponding angles and alternate interior angles are congruent.
4. Therefore, ∠AOB = ∠SOR.

(ii) O is the midpoint of BR:

1. Given that O is the midpoint of AS, we know that AO = OS.
2. Since AB is parallel to RS, we can conclude that ∠ABO and ∠RSO are corresponding angles.
3. By the corresponding angles theorem, we know that if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
4. Therefore, ∠ABO and ∠RSO are congruent.
5. Since AO = OS and ∠ABO = ∠RSO, we can apply the midpoint theorem.
6. The midpoint theorem states that if a line segment is cut by a midpoint, then the two resulting line segments are congruent.
7. Therefore, BO = OR.
8. Since BO = OR, we can conclude that O is the midpoint of BR.

Thus, we have shown that (i) ∠AOB = ∠SOR and (ii) O is the midpoint of BR.