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ABCDE is a regular pentagon. Sides AE and ED of the pentagon are extended as shown to form triangle EFG, which is right-angled at F. Vertices A and C are joined to form the triangle ABC. What is the value of sum (x° + y°)?
  • a)
    30°
  • b)
    45°
  • c)
    54°
  • d)
    60°
  • e)
    72°
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
ABCDE is a regular pentagon. Sides AE and ED of the pentagon are exten...
Step 1: Understand the question and draw inferences
To find –x°+ y°
Given – ABCDE is a regular pentagon.
This means, all sides of the pentagon are equal.
Therefore, AB = BC
In triangle ABC, two sides are equal.
Therefore, triangle ABC is an isosceles triangle.
Since angles opposite to equal sides are also equal,
Angle BAC = Angle ACB = x°
Let Angle ABC = i°
Therefore, by Angle Sum Property,
2x° + i° = 180°   .  .  . (1)
Now, considering the right triangle EFG,
Let angle FEG = e°
Therefore, by Angle Sum Property,
e° + y° + 90° = 180°
e° + y° = 90°    .  .  . (2)
Step 2: Finding the required values
We know that for interior angle of an n-sided regular polygon =
 
So, for n = 5, value of each interior angle =  
Therefore, i° = 108°    . . . (3)
Also, we know that Exterior Angle of an n-sided regular polygon = 360on
So, for n = 5, the value of each exterior angle = 360o/5 =  72°
Notice that Angle FEG is an exterior angle of the regular pentagon.
Therefore, e° = 72°   .  .  . (4)
Step 3: Calculating the final answer
From (1) and (3), we get:
2x° + 108° = 180°  
Solving this equation, we get x° = 36°
From (2) and (4), we get:
72° + y° = 90°
y° = 18°
Thus, x° + y° = 36° + 18° = 54°
Correct Choice -C
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ABCDE is a regular pentagon. Sides AE and ED of the pentagon are extended as shown to form triangle EFG, which is right-angled at F. Vertices A and C are joined to form the triangle ABC. What is the value of sum (x° + y°)?a)30°b)45°c)54°d)60°e)72°Correct answer is option 'C'. Can you explain this answer?
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