Q11 of 14 Page 77

If two straight lines intersect each other four angles are formed. Let’s prove that the sum of measurement of the four angles is four right angles.

Let us consider two straight lines PR and SQ intersecting at a common point O.


The question is just a twisted version of very simple question.


Proof:


We know that a right angle means that the angle is equal to 90°.


So, four right angles mean that the angle is equal to 4 × 90° which is equal to 360°.


Let ∠POQ = ∠1


∠QOR = ∠2


∠SOR = ∠3


∠POS = ∠4.


Now when we observe the above diagram, ray QO is perpendicular to line PR.


So ∠1 + ∠2 = 180°……… (i) [As they form linear pair that is sum of angles in a straight line is 180°]


Similarly ray RO is perpendicular to line SQ.


So ∠2 + ∠3 = 180°……… (ii) [As they form linear pair that is sum of angles in a straight line is 180°]


Now let us add equations (i) and (ii).


So, we get ∠1 + ∠2 + ∠3 + ∠4 = 180° + 180°


∠1 + ∠2 + ∠3 + ∠4 = 360°.


Hence all the angles sum up to four right angles that is 360°.


Hence it is proved that the sum of measurement of the four angles is four right angles.


More from this chapter

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9

Four rays meet at a point in such a way that measurement of opposite angles are equal. Let’s prove that two straight lines are formed by those four rays.

 10

Let’s prove that internal and external bisectors of an angle are perpendicular to each other.

 12

If triangle ∠PQR = ∠PRQ. If we extend QR on both sides, then two exterior angles are formed. Let’s prove that the measurement of external angles are equal.

 13

Two straight lines intersect each other at a point and thus four angles are formed. Let’s prove that the bisectors of these angles are two perpendicular straight lines.