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 P1 which also divides the segment in ratio m:n internally.
∴ AP/PB = m:n and AP1/P1B = m/n
∴ nAP = mPB and nAP1 =mP1B
nAP = m(AB - AP) and nAP1 = m( AB - AP1)
⇒ nAP = mAB - mAP and nAP1 = mAB - mAP1
⇒ (m+n) AP = mAB and (m + n)AP1 = mAB
⇒ AP = mAB/(m+n) and AP1 = mAB/(m+n)
∴ AP = AP1
∴ P and P1 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 AB2 ?
Answer: Since B lies between A and C,
∴ AB + BC = AC
BC = 3 and AC = 8
AB = AC - BC = 8 - 3 = 5
AB2 = 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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