Reasoning · Grade 4-2 Transformations and Angles
Problem
The crease is a mirror, so both sides match.
A fold is a mirror; a mirror never changes the size of an angle, so the crease always sits exactly halfway between an edge and where that edge lands.
8.G.A.1Create A Physical RepresentationThat fixes one base angle of the triangle.
A point sitting on a ray does not change where the ray points, so naming the angle by A instead of by E' names the same angle.
4.G.A.1Draw A DiagramThe parallel edges carry that angle across.
One straight line crossing two parallel lines repeats the same angles at both crossings, so the tilt of the crease is the same whether you read it at the bottom edge or at the top edge.
8.G.A.5Identify SubproblemsThe angle read off at the bottom edge is carried up to the top edge, because the two long edges of the strip are parallel.
The crease crosses both long edges, and one line crossing two parallel lines repeats the same angle at both crossings.
The fold itself never changes an angle's size, so the tilt of the crease is the same wherever it is measured.
Equal base angles mean two equal sides.
Two matching corner angles mean the triangle looks the same from the left as from the right, and a shape that is its own mirror image across a middle line has two equal sides.
4.G.A.2Identify SubproblemsSo it is always isosceles.
If the two base angles really are equal, folding the paper triangle along the line through the top corner should make the two slanted sides land exactly on each other.
8.G.A.5Create A Physical RepresentationFold a strip along a slanted crease and the crease angle shows up twice — once from the mirror and once from the parallel edges — so the little triangle it makes always has two equal sides.