Question
In a triangle, the sum of two angles is equal to the third angle. If the difference between these two angles is $20^\circ ,$ determine all the angles.
✓
Answer
Let the two angles of a triangle be $x$ and $y$ respectively.
Then, the $3^{rd}$ angle will be $180^\circ - (x + y).$
According to given information, we have
$x + y = 180^\circ - (x + y)$
$\Rightarrow 2(x + y) = 180^\circ$
$\Rightarrow x + y = 90^\circ \dots....(i)$
And,
$x + y = 20^\circ \dots....(ii)$
Adding eqns. $(i)$ and $(ii),$ we have
$2x = 110^\circ$
$\Rightarrow x = 55^\circ$
$\Rightarrow 55^\circ + y = 90^\circ$
$\Rightarrow y = 35^\circ$
$\Rightarrow 3^{rd}$ angle
$= 180^\circ - (55^\circ + 35^\circ )$
$= 180^\circ - 90^\circ$
$= 90^\circ$
Hence, the three angles of a triangle are $55^\circ , 35^\circ$ and $90^\circ .$
Then, the $3^{rd}$ angle will be $180^\circ - (x + y).$
According to given information, we have
$x + y = 180^\circ - (x + y)$
$\Rightarrow 2(x + y) = 180^\circ$
$\Rightarrow x + y = 90^\circ \dots....(i)$
And,
$x + y = 20^\circ \dots....(ii)$
Adding eqns. $(i)$ and $(ii),$ we have
$2x = 110^\circ$
$\Rightarrow x = 55^\circ$
$\Rightarrow 55^\circ + y = 90^\circ$
$\Rightarrow y = 35^\circ$
$\Rightarrow 3^{rd}$ angle
$= 180^\circ - (55^\circ + 35^\circ )$
$= 180^\circ - 90^\circ$
$= 90^\circ$
Hence, the three angles of a triangle are $55^\circ , 35^\circ$ and $90^\circ .$
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