Angles are shown in the given figure. What is value of ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8? 
360°
Concept Used
- Vertically opposite angles formed by two intersecting lines are congruent.
- The sum of the angles of a triangle is 180°.
Calculation
From the given figure, we can write:
⇒ ∠BAD = 180° – (∠1 + ∠2)
⇒ ∠ABC = 180° – (∠3 + ∠4)
⇒ ∠BCD = 180° – (∠5 + ∠6)
⇒ ∠ADC = 180° – (∠7 + ∠8)
Adding all four equations, we get:
⇒ ∠BAD + ∠ABC + ∠BCD + ∠ADC = 720° – (∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8)
∵ The sum of angles in a quadrilateral = 360°,
⇒ 360° = 720° – (∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8)
∴ (∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8) = 720° – 360° = 360°
Alternate Method
This is a star-and-mesh type problem. The types of triangles are not mentioned.
Assume all triangles are isosceles and the figure between the triangles is a square. Then, each triangle will be isosceles and right-angled.
In any such triangle:
90 + 2∠1 = 180
∠1 = 45°
Now, ∠1 = ∠2 = ∠3 = ∠4 = ∠5 = ∠6 = ∠7 = ∠8 = 45°
∴ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 = 8 × 45° = 360°
∴ The required sum is 360°.
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