CPD is twice as big as APB, and reflex angle is five times as big as APB. Make an equation in x, solve the equation, and find the fair angles.
Answer (1)
To solve this problem, let's first define what is given and what is being asked.
Let's denote:
∠ A PB = x
∠ CP D = 2 x
The reflex angle (let's denote it as ∠ R ) is five times ∠ A PB , so ∠ R = 5 x .
A reflex angle is greater than 180 degrees but less than 360 degrees. To find the relationship between these angles, we need to consider that the total angle around point P is 360 degrees.
The sum of ∠ A PB , ∠ CP D , and the reflex angle ∠ R equals 360 degrees, based on the full circle around a point:
x + 2 x + 5 x = 360
Simplify the equation:
8 x = 360
Now, solve for x by dividing both sides of the equation by 8:
x = 8 360 = 45
So, we have:
∠ A PB = 4 5 ∘
∠ CP D = 2 × 4 5 ∘ = 9 0 ∘
The Reflex Angle ∠ R = 5 × 4 5 ∘ = 22 5 ∘
These are the fair angle measures based on the given relationships.
