CBSE Class 9 - Maths - Ch6 - Lines and Angles (Set-2)

Octahedron

Lines and Angles: Q & A

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°