Problem
Prove triangle angle sum with a parallel line
A straight line DE runs across the top of the picture. The apex A of triangle ABC sits on that line. The base BC is parallel to line DE. Show that the triangle's three angles add to 180 degrees.
Reasoning · Grade 4-2 Parallel Lines and Angles
Your answer
Watch out Measuring and adding is tempting, but the task is to show why.
How to solve
Strategy Draw a Diagram — The three angles of the triangle sit at three different corners, far apart, so there is no way to see their total. The parallel line drawn through A is the tool that fixes this: it lets me carry a copy of the angle at B and a copy of the angle at C up to the vertex A. Once all three angles are gathered at the single point A, side by side along a straight line, their total is obvious. So the job splits into two small subproblems — move ∠B up, move ∠C up — followed by one easy observation about a straight line.
1STEP 1
Name the three angles I want to add
At A three angles already lie along a straight line.
∠ BAC + ∠ ABC + ∠ ACB = 180° ?
Writing down exactly which three angles are meant, and noticing that the only extra line in the picture is the parallel one, is what points the whole argument in the right direction.
4.G.A.1Draw A Diagram2STEP 2
Notice the three angles that already sit on the line at A
Those three add to 180 degrees.
∠ DAB + ∠ BAC + ∠ CAE = 180°
A straight line is a half turn, which is 180°, and angles that sit side by side just add up — so any group of angles that fills a straight line at one point must total 180°.
7.G.B.5Identify Subproblems3STEP 3
Slide the angle at B up to A along the side AB
Along AB, the angle at B equals the left one.
∠ DAB = ∠ ABC
Parallel lines never lean towards or away from each other, so any straight line meets both of them at exactly the same slant — which is why the matching angles at the two crossings must be the same size.
8.G.A.5Organize Information In More WaysThe angle at B can be slid up to A along the side AB, because the helper line at A is parallel to BC.
Why?
The side AB crosses two parallel lines, so the angle it makes with one is repeated where it meets the other.
🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.
Why?
Once all three angles sit side by side along the line at A, they fill that straight line and so total 180 degrees.
🧱A straight angle is 180 degreesAngles sitting side by side along a straight line always fill up 180 degrees.
4STEP 4
Slide the angle at C up to A along the side AC
Along AC, the angle at C equals the right one.
∠ CAE = ∠ ACB
The same parallel-line fact is used a second time, mirrored left to right — one line of reasoning, applied to each of the two slanted sides in turn.
8.G.A.5Organize Information In More Ways5STEP 5
Swap the copies in and read off 180°
Swapping them in makes the triangle's angles 180 degrees.
∠ ABC + ∠ BAC + ∠ ACB = ∠ DAB + ∠ BAC + ∠ CAE = 180°
Swapping an angle for another angle of exactly the same size cannot change a total, so the 180° that belonged to the straight line at A now belongs to the three corners of the triangle.
4.MD.C.7Identify SubproblemsAnswer
180 degrees
The argument never used a single measurement, so it holds for a triangle of any shape — which is right, because the 180° fact is supposed to be true for every triangle. It also survives a direct measurement check on the printed figure: a protractor on the drawing gives roughly 80° at A, 46° at B and 54° at C, and 80° + 46° + 54° = 180°. One more sanity check on the logic: the parallel line DE was essential, since it is the only thing that lets an angle down at B be copied up to A; without it the three corners could never be brought together at one point.
Takeaway
A parallel line through the top corner lets you slide the other two angles up to meet it — and three angles stacked along a straight line always make 180°.
- Name the three angles I want to add
- Notice the three angles that already sit on the line at A
- Slide the angle at B up to A along the side AB
- Slide the angle at C up to A along the side AC
- Swap the copies in and read off 180°