Problem
Why a parallelogram's opposite angles are equal
A parallelogram ABCD is drawn. Two opposite angles are marked with arcs. Not a single measurement is given. Show why those two angles must be equal.
Reasoning · Grade 4-2 Polygons and Angles
Your answer
Watch out Measuring both and calling them equal is tempting, but measuring is not a proof.
How to solve
Strategy Draw a Diagram — The two angles to be compared sit at opposite corners, as far apart as they can be, so there is no single crossing that links them. Adding one line to the picture fixes that: extend side BC past C to a new point E. That extension creates a third angle at C, outside the parallelogram, which is a next-door neighbour of both of the original angles in the parallel-line sense. Instead of comparing angle ABC with angle ADC directly, compare each of them with that go-between angle — two easy subproblems in place of one impossible one.
1STEP 1
Write down what "parallelogram" actually promises
A parallelogram promises opposite sides are parallel.
AD ∥ BC and AB ∥ DC
Starting from the definition instead of from the picture is what makes the argument work for every parallelogram, not just the one that happens to be drawn.
4.G.A.2Organize Information In More Ways2STEP 2
Extend the bottom side to make a go-between angle
Extend the bottom side to make a go-between angle.
B, C, E lie on one straight line
Extending a side costs nothing and turns two of the parallelogram's sides into full lines, which is what the parallel-line angle facts need.
4.G.A.2Draw A Diagram3STEP 3
First subproblem: angle ABC = angle DCE
One pair of parallels makes it equal to angle ABC.
∠ ABC = ∠ DCE (AB ∥ DC)
Sliding the whole crossing picture along the line from B to C lands it exactly on the second parallel, so every angle lands on a copy of itself.
8.G.A.5Identify SubproblemsExtending the bottom side creates a go-between angle that equals the angle at B, because two sides of the shape are parallel.
Why?
The extended line crosses two parallel sides, so the angles it makes with them are equal.
🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.
Why?
If the go-between angle equals the first angle and also equals the second, the first and the second must equal each other.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
4STEP 4
Second subproblem: angle ADC = angle DCE
The other pair makes it equal to angle ADC.
∠ ADC = ∠ DCE (AD ∥ BC)
Turning the crossing picture halfway round about the middle of DC maps one parallel onto the other, so the two inside angles on opposite sides must match.
8.G.A.5Identify Subproblems5STEP 5
Put the two subproblems together
Linking them shows the opposite angles are equal.
∠ ABC = ∠ DCE = ∠ ADC → ∠ ABC = ∠ ADC
Comparing two unknown things to one shared middle thing is a move any young solver already uses with lengths — "my stick matches the door and your stick matches the door, so our sticks match".
8.G.A.5Organize Information In More WaysAnswer
the two angles are equal
The explanation uses nothing but the definition of a parallelogram, so it cannot depend on the particular drawing — tilt the parallelogram more or less and every step still reads the same. It also passes the special cases: in a rectangle all four angles are 90°, so opposite ones are certainly equal; in a very flat, strongly leaning parallelogram the two acute corners are opposite each other and the two obtuse corners are opposite each other, which is exactly what the argument predicts. And it is consistent with the other parallelogram fact, that neighbouring angles add to 180°: if angle ABC + angle BCD = 180° and the opposite angles were unequal, the four angles could not total 360°.
Takeaway
If two angles you cannot measure are each equal to the same third angle, they have to be equal to each other — that is the whole reason a parallelogram's opposite corners match.
- Write down what "parallelogram" actually promises
- Extend the bottom side to make a go-between angle
- First subproblem: angle ABC = angle DCE
- Second subproblem: angle ADC = angle DCE
- Put the two subproblems together