Question : In $\triangle ABC, \angle A+\angle B=145^{\circ}$ and $\angle C+2\angle B=180^{\circ}$. State which one of the following relations is true.
Option 1: CA = AB
Option 2: CA < AB
Option 3: BC > AB
Option 4: CA > AB
Correct Answer: CA > AB
Solution :
In $\triangle$ABC,
$\angle$ A + $\angle$ B + $\angle$ C = 180° [Angle sum property of a triangle] _____(1)
$\angle$ A + $\angle$ B = 135° _____(2)
$\angle$ C + 2$\angle$ B = 180° _____(3)
From equation (1) and (3)
$\angle$ A + $\angle$ B + $\angle$ C = $\angle$ C + 2$\angle$ B
⇒ $\angle$ A = $\angle$ B = 67.5° (since $\angle$ A + $\angle$ B = 135°)
⇒ $\angle$ C = 180° – 135° = 45°
⇒ $\angle$B > $\angle$C
⇒ CA > AB
Hence, the correct answer is CA > AB.
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