Folding makes congruent corresponding angles

4.G.A.1 · adapt · grade 4

Archetype: Transformations Preserve Measures · step in a 8-type progression

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge, labeled ⓑ, measures $25^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

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Understand

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.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle b between the folded edge and the bottom edge is 25 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The bottom edge of the paper is a straight line (180 deg) at N.

Plan

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

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.

Execute

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the angle the original edge made and the angle the folded edge makes are mirror images. The folded edge makes 25 deg with the bottom edge, and by the reflection the crease's other side gives another equal 25 deg slant.

\text{both base slants} = 25^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

#7 Identify Subproblems 4.MD.C.7

Along the bottom edge at N the angle is a straight 180 deg. The two equal 25 deg slants of the standing flap sit on that straight line, and the peak angle a together with them must complete the straight angle around the fold.

a = 180^\circ - 25^\circ - 25^\circ

The folded flap is an isosceles triangle with two equal base angles of 25 deg, so the top angle is the rest of the 180 deg triangle total.

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 50^\circ = 130^\circ

Two small 25 deg corners leave a wide 130 deg angle at the top of the thin folded flap.

Answer: 130 degrees

Review

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.

Use Draw a Diagram (tool 1): mark the equal fold angles as 25 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(25) = 130 deg.

Standards · min grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at N.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 25 deg base angles from 180 deg to find the peak angle.

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