The triangle that lost a corner
Pip folds three corners into one straight queue. Nobody gets to jump the line.
The formula in this story
Story 6 of 8
What you will take away
Explain why three interior angles fill a straight angle on a flat plane.
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What do the words mean?
An angle measures a turn. A straight angle is 180°. A triangle here joins three noncollinear points by straight sides on an ordinary flat (Euclidean) plane.
Parallel lines do not meet, however far extended. A crossing line makes alternate interior angles: inside the pair of lines, on opposite sides of the crossing line. For parallel lines, these angles are equal. We use this earlier geometry fact.
No other case needs to be read first.
A suspicious plan
Pip cuts a paper triangle into three corners. “If one corner is missing, can I charge less for the proof?”
Draw the line through A parallel to BC. This is a fact about a flat Euclidean plane.
The clue
Mika keeps the triangle intact and draws a line through its top vertex, parallel to the opposite side.
Copy β and γ beside A using equal alternate interior angles.
The twist
The two base angles reappear beside the top angle as alternate angles. Together they fill a straight line. Pip finds his lost corner under his hat.
The copied angles and α fill one straight angle: 180°. The fan below records those same three sizes.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Through vertex A, draw a line parallel to side BC. We use Euclidean parallel-line properties.
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
In a Euclidean plane, the interior angles of a nondegenerate triangle sum to 180°.
- Through vertex A, draw a line parallel to side BC. We use Euclidean parallel-line properties.
- The angle at B equals the alternate angle between BA and the parallel line. The angle at C likewise moves to the other side of A.
- These two angles and the angle at A partition a straight angle, so their sum is 180°. A globe is not a Euclidean plane; this proof is not a claim about every surface.
Pip has one more idea…
Two angles are 50° and 60°. How large is the third?
Read the explanation for 70°
Yes: 180 − 50 − 60 = 70. All three angles together use the straight-angle budget.
Read the explanation for 110°
110° is the sum of the two known angles. Subtract it from 180° to get 70°.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
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These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.