Reasoning · Grade 5-2 Congruent Figures
Problem
Note where each part sits.
The angles of a triangle always total 180 degrees, so this quick sum confirms the figure has been read correctly before any description is tested.
8.G.A.5Make A Systematic ListPinning a triangle needs at least one side.
Each package is really a set of building instructions - lay the known side down, then the rest of the triangle has no choice left - and that is exactly what 'must be congruent' means.
7.G.A.2Make A Systematic ListA description pins a triangle down only if any two triangles matching it can be laid exactly on top of each other.
Sliding, turning or flipping keeps every length and angle, so two triangles are the same exactly when one such move carries one onto the other.
Three angles alone cannot pin a triangle down, because the angles already add to one straight angle whatever the size of the triangle.
(1) gives only two sides, so out.
Two sticks joined at one end swing freely, and drawing the wide-open version once is enough to knock the description out.
7.G.A.2Eliminate Possibilities(2) gives two sides and the included angle: keep.
The word 'included' is doing all the work: it is the difference between a hinge that can swing and a hinge that has been locked at 30 degrees.
7.G.A.2Eliminate Possibilities(3) gives three sides: keep.
A triangle made of three fixed rods is rigid - it cannot be squashed the way a four-sided frame can - so the three lengths leave no freedom at all.
7.G.A.2Eliminate Possibilities(4) gives three angles only, so out.
Equal angles only guarantee the same shape; a photocopy enlarged to 200 percent has all the same angles, which is exactly why size has to be stated somewhere.
8.G.A.5Eliminate Possibilities(5) gives a side and its two angles: keep.
Two straight arms set off at fixed angles from the two ends of a fixed base cross at exactly one point, so the drawing can only come out one way.
7.G.A.2Draw A DiagramThe survivors are 2, 3 and 5.
Once each of the five has been kept or eliminated for a stated reason, nothing is left to check - a finite list has been worked all the way through.
7.G.A.2Eliminate PossibilitiesThree sides, two sides with the angle between them, or one side with the angles at its ends - anything less and the triangle can still change.