Question

The angles of a triangle are 3x^0, (2x+20)^0 and (5x-40)^0.find the angles hence show that the triangle is an equilateral triangle?​

Answer 1

Answer:

The angles of a triangle are 3x, (2x+20), and (5x-40).Find the angles. Show that the triangle is an equilateral triangle."

Angles of a triangle add up to 180.

3x + 2x + 20 + 5x - 40 = 180 {sum of angles equals 180}

10x - 20 = 180 {combine like terms}

10x = 200 {added 20 to both sides}

x = 20 {divided both sides by 10}

Substitute 20, in for x, into sides

3x = 60

2x + 20 = 60

5x - 40 = 60

It is an equilateral triangle because all of the angles are equal}

Answer 2

$\large\underline\blue{\bold{Given :-  }}$

$\begin{gathered}\begin{gathered}\bf The \: angles \: of \: triangle \: are - \begin{cases} &\sf{3x°} \\ &\sf{(2x + 20)°}\\ &\sf{(5x - 40)°} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf To \:- \begin{cases} &\sf{find \: the \: angles \: of \: triangle} \\ &\sf{show \: triangle \: is \: equilateral} \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{ \tt \: Solution :- }}$

$\begin{gathered}\begin{gathered}\tt The \: angles \: of \: triangle \: are - \begin{cases} &\sf{3x°} \\ &\sf{(2x + 20)°}\\ &\sf{(5x - 40)°} \end{cases}\end{gathered}\end{gathered}$

$\tt \: ☆ \: We \: know \: the \: angle \: sum \: property \: of \: triangle \\ \tt \: that \: sum \: of \: angles \: of \: a \: triangle \: is \: 180° \: \: \: \: \: \: \:$

$\tt \: \therefore \: 3x + 2x + 20 + 5x - 40 = 180$

$\tt\implies \:10x - 20 = 180$

$\tt\implies \:10x = 200$

$\tt\implies \:x = 20$

$\begin{gathered}\begin{gathered}\bf Hence, \: angles \: are - \begin{cases} &\sf{3x° = 3 \times 20 = 60°} \\ &\sf{(2x + 20)° = 2 \times 20 + 20 = 60°}\\ &\sf{(5x - 40)° = 5 \times 20 - 40 = 60°} \end{cases}\end{gathered}\end{gathered}$

$\tt \: \sf \:  ⟼Since, \: each \: angle \: is \: 60° \\ \tt\implies \:Triangle \: is \: equilateral.$

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Properties of a triangle

A triangle has three sides, three angles, and three vertices.

The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

The sum of the length of any two sides of a triangle is greater than the length of the third side.

The side opposite to the largest angle of a triangle is the largest side.

Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.

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