Folded angles are equal; chain to the unknown

4.MD.C.7 · take · grade 4

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

A triangular sheet of paper is folded as shown so that side $BE$ and side $DE$ have the same length. Find the measure of angle $EFC$ .

Show solution

Understand

In triangle BDC (B and C on the base, D the apex), the base angle at B is 40 degrees and at C is 80 degrees. The paper is folded so that BE and DE are equal, and the apex flap lands at point F on the right slant side DC. I must find the angle EFC.

Givens

Triangle BDC has angle B = 40 degrees and angle C = 80 degrees
E lies on base BC with BE = DE
The paper is folded along a crease so the apex flap (containing D) comes down and F lands on slant side DC
F lies on line DC, so D, F, C are along the same slant side

Unknowns

The measure of angle EFC

Constraints

Angles in a triangle add to 180 degrees
Folding preserves angle sizes (a folded angle equals its original)
An isosceles triangle has two equal base angles

Plan

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

Folding paper is a hands-on action, so picturing (or actually doing) the fold shows that folded angles stay equal. Then I break the figure into small triangles and chain known angles step by step until I reach angle EFC.

Execute

#7 Identify Subproblems 4.MD.C.7

In triangle BDC the angles add to 180 degrees, so the apex angle at D is 180 - 40 - 80 = 60 degrees.

\angle BDC = 180^\circ - 40^\circ - 80^\circ = 60^\circ

Grade 4 students know the three angles of a triangle always total 180 degrees.

#7 Identify Subproblems 4.MD.C.7

Since BE = DE, triangle BED is isosceles, so its base angles are equal: angle BDE = angle DBE = 40 degrees. Then angle DEB = 180 - 40 - 40 = 100 degrees, and since B, E, C are in a line, angle DEC = 180 - 100 = 80 degrees.

\angle DEB = 180^\circ - 40^\circ - 40^\circ = 100^\circ,\quad \angle DEC = 180^\circ - 100^\circ = 80^\circ

Equal sides give equal base angles, and angles on a straight line add to 180 degrees.

#10 Create a Physical Representation 4.MD.C.7

Folding does not change angle sizes, so the crease through E reflects the apex flap onto F. Tracking the preserved angles, triangle EFC ends up with the base angle at C unchanged (angle FCE = 80 degrees) and the angle at E equal to 40 degrees (the reflected copy of the 40-degree piece).

\angle FCE = 80^\circ,\quad \angle FEC = 40^\circ

A folded shape lies exactly on top of its original, so matching angles stay the same size.

#7 Identify Subproblems 4.MD.C.7

In triangle EFC the three angles add to 180 degrees: angle EFC = 180 - 80 - 40 = 60 degrees.

\angle EFC = 180^\circ - 80^\circ - 40^\circ = 60^\circ

Once two angles of a triangle are known, the third comes from the 180-degree total.

Answer: 60 degrees

Review

Angle EFC = 60 degrees is acute and matches the apex angle of the original triangle, which is reasonable for a fold that brings the 60-degree apex region down onto the slant side. The triangle EFC checks out: 80 + 60 + 40 = 180 degrees.

Draw the diagram to scale (tool 1) and measure angle EFC with a protractor as a confirmation of the 60-degree result reached by angle chasing.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining angles through the triangle-sum, isosceles base angles, straight-line angles, and the equal folded angle to reach angle EFC.

💡 This only needs Grade 4 angle-adding and the fact that folded angles stay equal!