Geometry & Figures

Problem

Folding makes congruent corresponding angles

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge (call it b) is 25 deg. I need angle a at the peak M, where the two folded faces meet.
25° a M N D C b
Geometry
Your answer
degrees
How to solve
Strategy Create a Physical Representation — Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle with N. The 25 deg slant at N appears on both sides of the fold, and the peak angle is what is left of a straight angle after removing those two equal base pieces.
1STEP 1

The fold makes two equal angles at N

The fold is a mirror, so the 25 deg slant at N is copied to another 25 deg on the crease's far side.

both base slants = 25° each
2STEP 2

Use the straight bottom edge

The bottom edge at N is a straight 180 deg, and the two 25 deg base slants plus the peak angle a fill that straight line.

a = 180° - 25° - 25°
3STEP 3

Compute the peak angle

Subtract the two equal base angles from 180 deg.

a = 180° - 50° = 130°
Answer
130 degrees
a = 180° - 25° - 25° = 130°
Because each base angle is only 25 deg, the flap is thin and tall, so a wide (obtuse) peak angle near 130 deg makes sense. Check: 25 + 25 + 130 = 180 deg, exactly a triangle's angle sum.
Takeaway

A fold is a mirror, so the two base angles match at 25 deg each; the peak is just 180 deg minus those two!

  • The fold makes two equal angles at N
  • Use the straight bottom edge
  • Compute the peak angle
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