Question
In $\triangle\text{ABC},$ if $\angle\text{A}=\angle\text{C},$ and exterior angle $ABX = 140^\circ ,$ then find the angles of the triangle.
✓
Answer
Given, $\angle\text{A}=\angle\text{C}$ and exterior $\angle\text{ABX}=140^{\circ}$
Let $\angle\text{A}=\angle\text{C}=\text{x}$
According to the exterior angle property,
$\text{Exterior}\angle\text{B}=\text{interior}\angle\text{A}+\text{interior}\angle\text{C}$
$\Rightarrow140^{\circ}=\text{x}+\text{x}$
$\Rightarrow140^{\circ}=\text{2x}$
$\Rightarrow\text{x}=\frac{140^{\circ}}{2}=70^{\circ}$
So, $\angle\text{A}=\angle\text{C}=70^{\circ}$
Now, $\Rightarrow\angle\text{A}+\angle\text{B}+\angle\text{C}=180^{\circ} $ [angle sum property of a triangle]
$\Rightarrow\angle\text{70}^{\circ}+\angle\text{B}+70^{\circ}=180^{\circ}$
$\Rightarrow\angle\text{B}+140^{\circ}=180^{\circ}$
$\Rightarrow\angle\text{B}=180^{\circ}-140^{\circ}$
$\Rightarrow\angle\text{B}=40^{\circ}$
Hence, all the angles of the triangle are $70^\circ , 40^\circ $ and $70^\circ .$
Let $\angle\text{A}=\angle\text{C}=\text{x}$
According to the exterior angle property,
$\text{Exterior}\angle\text{B}=\text{interior}\angle\text{A}+\text{interior}\angle\text{C}$
$\Rightarrow140^{\circ}=\text{x}+\text{x}$
$\Rightarrow140^{\circ}=\text{2x}$
$\Rightarrow\text{x}=\frac{140^{\circ}}{2}=70^{\circ}$
So, $\angle\text{A}=\angle\text{C}=70^{\circ}$
Now, $\Rightarrow\angle\text{A}+\angle\text{B}+\angle\text{C}=180^{\circ} $ [angle sum property of a triangle]
$\Rightarrow\angle\text{70}^{\circ}+\angle\text{B}+70^{\circ}=180^{\circ}$
$\Rightarrow\angle\text{B}+140^{\circ}=180^{\circ}$
$\Rightarrow\angle\text{B}=180^{\circ}-140^{\circ}$
$\Rightarrow\angle\text{B}=40^{\circ}$
Hence, all the angles of the triangle are $70^\circ , 40^\circ $ and $70^\circ .$
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