How many degrees is angle x?
A hard-level "Angle" problem.A regular pentagon is built out of just 108° and 36°. Remember that every diagonal produces a 36° isosceles triangle and these problems go fast.
GivenTwo diagonals drawn in a regular pentagon
Hints
- First, one angle of the pentagon. A pentagon splits into three triangles, so its angles total 180 × 3 = 540°. All five are equal, so each is 540 ÷ 5 = 108°.
- Tidy the lines and look only at triangle ABC. AB and BC are both sides of the pentagon, so they are equal: an isosceles triangle with a 108° apex.
- Being isosceles, its two base angles are equal: (180 − 108) ÷ 2 = 36°. A pentagon's diagonal always cuts off 36°.
- The other diagonal BE does exactly the same. Triangle ABE is isosceles too, so that angle is 36° as well.
- Look at triangle ABP formed by the two diagonals: both of its lower angles are 36°. So angle x is 180 − 36 − 36.
Answer 108°
More "Angle" problems
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- What is the angle at the centre of the sector?