Question

The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4. What is the measure of the four angles?

Answer 1

$\fbox\red{αвѕωєя}$

hínt: tσ ѕσlvє thíѕ quєѕtíσn wє ѕhσuld knσw thαt thє ѕum σf αll thє íntєríσr αnglєѕ σf thє quαdrílαtєrαl íѕ 360 dєgrєєѕ.

cσmplєtє ѕtєp вч ѕtєp ѕσlutíσn:

ín thє quєѕtíσn, wє hαvє tσ fínd thє αnglєѕ σf α quαdrílαtєrαl thαt αrє ín thє rαtíσ 1: 2: 3: 4.

nσw, ít íѕ knσwn thαt íf thє rαtíσ íѕ gívєn αѕ α : в : c : d, thєn thє αctuαl numвєrѕ αrє αх, вх, cх, dх, whєrє х íѕ thє cσmmσn fαctσr σf αll thє numвєrѕ.

ѕσ, nσw thє αnglєѕ thαt αrє ín rαtíσ σf 1: 2: 3: 4, cαn вє wríttєn αѕ х, 2х, 3х αnd 4х ín dєgrєєѕ.

nσw, ít íѕ vєrч wєll knσwn thαt thє ѕum σf αll thє íntєríσr αnglєѕ σf thє quαdrílαtєrαl íѕ 360 dєgrєєѕ.

ѕσ, thє αnglєѕ х, 2х, 3х αnd 4х ín dєgrєєѕ αrє αdd αѕ fσllσwѕ:

⇒х+2х+3х+4х=360∘⇒10х=360∘⇒х=36∘⇒х+2х+3х+4х=360∘⇒10х=360∘⇒х=36∘

ѕσ hєrє thє αnglєѕ wíll вє х=36∘х=36∘, 2х=72∘2х=72∘, 3х=108∘3х=108∘, αnd 4х=144∘4х=144∘.

nσw wє cαn ѕєє thαt 2х+3х=72∘+108∘=180∘2х+3х=72∘+108∘=180∘ αnd х+4х=36∘+144∘=180∘х+4х=36∘+144∘=180∘

ѕσ thє αdjαcєnt pαír σf αnglєѕ íѕ 180 dєgrєєѕ. ѕσ thíѕ íѕ α trαpєzíum αѕ thє ѕum σf thє αdjαcєnt αnglєѕ σn thє pαrαllєl ѕídєѕ íѕ 180 dєgrєєѕ.

nσtє: trαpєzíum íѕ α kínd σf quαdrílαtєrαl. thє quαdrílαtєrαl íѕ α cσnvєх pσlчgσn whєrє αll íntєríσr αnglєѕ αrє lєѕѕ thαn 180 dєgrєєѕ. ít íѕ ímpσrtαnt thαt wє dσn’t gєt cσnfuѕєd wíth thє αdjαcєnt αnglєѕ αnd thє σppσѕítє αnglєѕ. íf thє σppσѕítє pαír σf αnglєѕ αrє thє ѕαmє fσr αnч quαdrílαtєrαl, thєn ít íѕ α pαrαllєlσgrαm αnd nσt trαpєzíum.

Answer 2

QuestioN :

The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4. What is the measure of the four angles?

GiveN :

• The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4.

To FiNd :

• The measure of the four angles.

ANswer :

The ratio of the angles of quadrilaterals = 1 : 2 : 3 : 4

Let the four angles of the quadrilateral be x, 2x, 3x, and 4x respectively.

The sum of four angles of a quadrilateral is 360°.

Therefore,

x + 2x + 3x + 4x = 360°

10x = 360°

x = 360°/10

x = 36°

Therefore,

First angle = x = 36°

Second angle = 2x = 2 × 36 = 72°

Third angle = 3x = 3 × 36 = 108°

Fourth angle = 4x = 4 × 36 = 144°

∴Hence, the measure of four angles is 36°, 72°, 108° and 144°.

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