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5. The measure of the second angle of a triangle is 20 more than the measure of the first, and the measure of the third angle is 10 less than three times the first. What is the measure of each of the three angles of a triangle?

Respuesta :

WindyMint

So first we have to find equation form angles.

Let :-

  • first angle of triangle be x

Part one :-

Statement :- The measure of the second angle of a triangle is 20 more than the measure of the first.

Equation :-

second angle = 20 + x

{ Remember people get's confused when to multiply and when to add. when ever there "more than" then we will add and when there is ,"times" we will multiply}

Part 2 :-

Statement :-

the third angle is 10 less than three times the first

Now just see first less means we have to subtract 10 from 3 times the first means 3 × first angle.

Equation :-

third angle = 3x - 10

Part 3 :-

Now let's represent :-

  • First angle be ∠1
  • Second angle be ∠2
  • third angle be ∠3

We know sum of angles of triangle is equal to 180°

So :-

  • ∠ 1 + ∠ 2 + ∠ 3 = 180°

  • x + 3x - 10 + 20 + x = 180°

  • 4x - 10 + 20 + x = 180°

  • 5x - 10 + 20 = 180°

  • 5x + 10 = 180°

  • 5x = 180° - 10°

  • 5x = 170

  • x = 170/5

  • x = 34°

Measure of angles :-

  • ∠1 = x
  • ∠ 1 = 34°

  • ∠2 = 20 + x
  • ∠2 = 20 + 34°
  • ∠2 = 54°

  • ∠3 = 3x - 10
  • ∠3 = 3 × 34 - 10
  • ∠3 = 102 - 10
  • ∠3 = 92°
Starrysoul100

[tex]\bold{\huge{\underline{ Solution }}}[/tex]

Given :-

  • We have given in the question that,
  • The measure of second angle is 20 more than the measure of first angle
  • The measure of third angle is 10 less than three times the first.

To Find :-

  • We have to find the measure of all the three angles of triangle.

Let's Begin :-

Let the first angle be x

According to the question,

  • Second angle = x + 20

[ 2nd angle is 20 more than the first angle ]

  • Third angle = 3x - 10

[ 3rd angle is 10 less than thrice times the first angle ]

By using Angle sum property

  • It states that the sum of the angles of triangles are equal to 180°

That is,

[tex]\sf{ x + ( x + 20) + ( 3x - 10) = 180{\degree}}[/tex]

[tex]\sf{ x + x + 20 + 3x - 10 = 180{\degree}}[/tex]

[tex]\sf{ 5x + 10 = 180{\degree}}[/tex]

[tex]\sf{ 5x = 180 - 10}[/tex]

[tex]\sf{ 5x = 170}[/tex]

[tex]\sf{ x = }{\sf{\dfrac{ 170}{5}}}[/tex]

[tex]\sf{ x = }{\sf{\cancel{\dfrac{ 170}{5}}}}[/tex]

[tex]\bold{ x = 34{\degree}}[/tex]

Thus,

  • The first angle is 34°

  • The second angle
  • [tex]\sf{ = 34 + 20}[/tex]
  • [tex]\bold{ = 54{\degree}}[/tex]

  • The third angle
  • [tex]\sf{ = 3(34 ) - 10}[/tex]
  • [tex]\sf{ = 102 - 10}[/tex]
  • [tex]\bold{ = 92 {\degree}}[/tex]

Hence, The three angles of the given triangle are 34° , 54° and 92°

[tex]\bold{\huge{\underline{ Verification }}}[/tex]

Here, we have

  • The first angle = 34°
  • The second angle = 54°
  • The third angle = 92°

By using Angle sum property :-

[tex]\sf{ 34 + 54 + 92 = 180{\degree}}[/tex]

[tex]\sf{ 88 + 92 = 180{\degree}}[/tex]

[tex]\sf{ 180 = 180{\degree}}[/tex]

[tex]\bold{ LHS = RHS }[/tex]

Hence, Verified