**Q1: Prove that two lines which are both parallel to the same line, are parallel to each other.**

Answer

**:****Given**: Three lines l, m, n in a plane such that l || m and m || n.

**To prove**: l || n

Proof: Suppose line l is not parallel to line n. Then l, n will intersect at some unique point, say at P.

⇒ P lies on l but does not lie on m, since l || m.

∴ Through point P outside m, there are two lines ( l and n ) and both are parallel to line m. This is not possible (violates Parallel Axiom).

∴ our assumption is wrong.

Hence l || n.

**Q2: Prove that, if P is a point which divides the line segment AB in the ratio m:n internally, then P is unique.**

Answer:

Suppose point P divides segment AB in the ratio m:n internally. Let us assume P is not unique.

⇒ There is another point P

_{1}which also divides the segment in ratio m:n internally.
∴ AP/PB = m:n and AP

_{1}/P_{1}B = m/n
∴ nAP = mPB and nAP

_{1}=mP_{1}B
nAP = m(AB - AP) and nAP

_{1}= m( AB - AP_{1})
⇒ nAP = mAB - mAP and nAP

_{1}= mAB - mAP_{1}
⇒ (m+n) AP = mAB and (m + n)AP

_{1}= mAB
⇒ AP = mAB/(m+n) and AP

_{1}= mAB/(m+n)
∴ AP = AP

_{1}
∴ P and P

_{1}are the same point, which contradicts our assumption.
Hence P is unique.

**Q3: How many least number of distinct points determine a unique line?**

Answer: Two.

**Q4: In how many point two distinct lines can intersect?**

Answer: One

**Q5: In how many points two distinct planes can intersect?**

Answer: Infinite number of points.

**Q6: If B lies between A and C and AC = 8, BC = 3. What is AB and AB**

^{2}?
Answer: Since B lies between A and C,

∴ AB + BC = AC

BC = 3 and AC = 8

AB = AC - BC = 8 - 3 =

**5**
AB

^{2 }= 5 ✕ 5 =**25****Q7: An angle is 14° more than its complement. Find the angle?**

Answer: Let the angle be p°.

Its complementary angle = 90° - p°

As per the given information,

p° = (90° - p°) + 14°

⇒ p° + p° = 90° + 14°

⇒ 2p° = 104°

⇒ **p° = 52°**

**Q8: What angle is equal to its supplement?**

Answer: 90°

Let the angle be y°.

∴ The other angle is = 180 - y

Since two angle are equal,

⇒ y = 180 - y

⇒ 2y = 180

⇒ y = 90°

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