Question

1 A transversal cuts two parallel lines at A and B. The two interior angles at A are bisected
and so are the two interior angles at B; the four bisectors form a quadrilateral ACBD.
Prove that
(1) ACBD is a rectangle.
(ii) CD is parallel to the original parallel lines.
Answer both the questions with proper explanation
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Answer 1

Answer:

1. Show that: ABCD is a square Given: Rectangle ABCD where AC bisects A, i.e. 1 = 2 & AC bisects C, i.e. 3 = 4 To prove: ABCD is a square Proof: A square is a rectangle when all sides are equal Now, AD BC & AC as transversal 1 = 4 Now, 1 = 2 & 1 = 4 Hence, 2 = 4 In ABC, 2 = 4 So, BC = AB But BC = AD & AB = DC From (1)
2. To prove → ABCD is a rectangle
3. AD, CD, AB, BC are bisectors of interior angles formed by transversal line with ∥ line.
4. ∠BCA=∠CAB
5. Hence,CB∥AB
6. Similarly,AB∥CB(∠CAB=∠ACB)
9. (Alternateangles)
12. Therefore quadrilateral ABCD is a ∥gram as both the pairs of opposite sides are ∥
13. ∠b+∠b+∠a+∠a=180
14. ∘
16. ⇒2(∠b+∠a)=180
17. ∘
19. ∠a+∠b=90
20. ∘
24. That is ABCD is ∥gram & one of the angle is ⊥ angle.
25. So, ABCD is a Rectangle.