Question : Three angles of a triangle are $(x-15^{\circ}),(x+45^{\circ}),$ and $(x+60^{\circ})$. Identify the type of triangle.
Option 1: Obtuse angle triangle
Option 2: Right angle triangle
Option 3: Isosceles triangle
Option 4: Equilateral triangle
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Correct Answer: Right angle triangle
Solution : Given: Angles of the triangles is $(x-15^{\circ}),(x+45^{\circ}),$ and $(x+60^{\circ})$. Let $\angle A =(x-15^{\circ}) $ $\angle B =(x+45^{\circ}) $ $\angle C =(x+60^{\circ}) $ We know that the sum of the angles of a triangle is $180^\circ$. So, $(x-15^{\circ})+(x+45^{\circ})+(x+60^{\circ}) = 180^{\circ}$ ⇒ $3x+90^{\circ}=180^{\circ}$ ⇒ $x=30^{\circ}$ Putting the values we get: $\angle A =(30-15^{\circ})=15^{\circ} $ $\angle B =(30+45^{\circ})=75^{\circ} $ $\angle C =(30+60^{\circ}) =90^{\circ}$ Since one of the angles is $90^{\circ}$, ⇒ The triangle is right-angled. Hence, the correct answer is the Right angle triangle.
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