Reasoning · Grade 4-2 Transformations and Angles
Problem
A turn leaves every angle unchanged.
Imagine the triangle cut out of card and pinned at A. Spinning the card cannot change any of its corners, so the only new number in the picture is the 30 degrees you spun it through.
8.G.A.1Visualize Spatial RelationshipsThe turn hands you a second triangle with exactly the same angles and side lengths as the first.
Rotating lays the triangle onto a copy of itself, so every length and every angle travels along unchanged.
The angle of the turn itself sits at the centre and can be added to the copied angles, since angle measures combine like lengths.
The triangle's third angle is 50 degrees.
Two angles of a triangle always give you the third for free, because the three of them together make a straight angle.
8.G.A.5Identify SubproblemsGive the crossing point a name.
A point sitting on a segment does not change the direction you look in, so an angle measured to P is the same as the angle measured to the far end of the segment - that is the trick that turns two facts about the whole picture into two angles of one small triangle.
4.G.A.1Draw A DiagramIn the small triangle that angle is 80 degrees.
Once two of a triangle's three angles are known, the third is just what is left of 180 degrees - no measuring needed.
8.G.A.5Identify SubproblemsUsing the straight line gives 100 degrees.
Standing at the crossing and swinging your arm from one side of the straight line to the other is half a full turn, which is 180 degrees - so the two angles on the same side of AB must fill that 180 between them.
7.G.B.5Identify SubproblemsChasing the other way also gives 100 degrees.
Two independent routes landing on the same number is the best evidence that no angle was misread off the figure.
4.MD.C.7Identify SubproblemsName the point the picture forgot to name, and the messy overlap turns into one small triangle whose angles you already know.