Why the three angles of any triangle add to 180°

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

Constructing from two angles and the side between them — that fixing two angles fixes the shape, and the observation that the third angle does not change when the base does
The straight angle: a straight line makes 180° at any point on it
Chapter 5 (Part I): parallel lines cut by a transversal, and the equality of alternate angles
Adding and subtracting angle measures in degrees
The four angles at a crossing: vertically opposite and linear pairs — what it takes for an argument to count as a proof rather than as evidence

What they should be able to do

Given two angles of a triangle, find the third by adding a parallel line and arguing, not by measuring
Identify the two pairs of alternate angles created when a line through the apex runs parallel to the base
Reproduce the general argument for the angle sum property using letters rather than chosen values
State the angle sum property and say what makes it a property of every triangle
Explain what an auxiliary line contributes when it carries no new measurement
Verify the property by folding a paper triangle, and say what the folding demonstrates and what it does not
Define an exterior angle of a triangle and locate one in a labelled figure
Compute an exterior angle from two given interior angles, in two steps
Notice the relation between an exterior angle and the two interior angles at the other vertices, and say how it follows

Where it usually goes wrong

"It adds to 180° because I measured lots of triangles and it did." Every measurement is approximate, and a hundred approximations do not make a certainty. The chapter's own reflection on p.166 is about precisely this — the relationship was invisible until a line was added, and then it was forced.
"The parallel line is part of the triangle." It is not. Nothing in the triangle changes when XY is drawn, and no length on XY is ever used. It exists so that three angles that were scattered can be seen sitting on one line.
"A very large triangle would have a bigger angle sum." Size is irrelevant; the argument never mentions a length. This is the same lesson as the previous topic's "the base does not matter", arriving one step further on.
"The paper fold is the proof." It is a check. Folding shows the three angles fitting along a straight edge for that triangle, cut with scissors and folded by hand. The chapter separates the two on the same page — it uses the language of proof for the lettered argument and the language of verifying for the fold.
"A triangle can have two right angles if it is big enough." Two right angles already use the whole 180°, leaving nothing for the third. Note that this is the same fact the previous topic reached from the parallel-line side, now available as a one-line consequence.
"The exterior angle is outside, so it is the angle between the two extended sides." It is measured between one extended side and the side next to it at the same vertex, and it shares an arm with the interior angle there. Reading it off the wrong pair of arms is the standard error, and the figure on p.167 is what fixes it.
"Euclid proved it, so it is true." The chapter's point is the opposite way round: the argument is what makes it true, and it is credited to The Elements because that is where this particular idea is found.

Questions to check understanding

Given two angles of a triangle, find the third and justify it with the parallel line (asked in exactly this form on p.165)
Given a triangle with two equal angles and the third stated, find the equal pair
Say whether a triangle can have two right angles, or two obtuse angles, and why
Find an exterior angle from two given interior angles, showing both steps
State what all three angles must be if they are equal
Explain what the added parallel line contributes to the argument
Distinguish a check by measurement from a proof, using the two treatments on p.166
Board-style items combine this with the constructions of §7.3: construct, then compute a missing angle without measuring it

Examples worth working on the board

The setup question (Part I, §7.3, p.164). Inputs: two angles, 60° and 70°, and an included side of 5 cm, then the same two angles on a 7 cm side. The chapter has the student measure the third angle in both. The observation — no change — is the reason for everything that follows.
Fig. 7.6 (Part I, §7.3, p.164). Checked against the printed page. ∆ABC with B at the lower left, C at the lower right and A at the top; 50° arced at B and 70° at C. A horizontal line is drawn through A, its left end labelled X and its right end labelled Y. There is a small arc at A between the two triangle sides. Inputs: the two given angles, and that XY runs parallel to BC.
The alternate-angle step (Part I, §7.3, p.164). Inputs: XY is parallel to BC; AB cuts both, so ∠XAB pairs with ∠B; AC cuts both, so ∠YAC pairs with ∠C.
The arithmetic at A (Part I, §7.3, p.165). Inputs: 50°, 70°, and that the three angles at A together make a straight angle. The 60° is the output.
The four angle pairs to complete (Part I, §7.3, p.165). 36° and 72° · 150° and 15° · 90° and 30° · 75° and 45°. The second is worth pausing on — a 150° angle leaves very little for the other two.
The equal-angles question (Part I, §7.3, p.165). Two questions in one: what the third angle is when two are 70°, and what all three must be if they are all equal. Both are inputs; both are the student's to answer.
The isosceles-style Try This (Part I, §7.3, p.165). Checked against the printed page. A tall triangle with A at the apex carrying 50°, and small arcs at B and C showing that those two angles are marked as equal. No values are printed at B and C. Inputs: 50° at the apex, and that the other two are equal.
Fig. 7.7 and the general argument (Part I, §7.3, p.166). Checked against the printed page. The same configuration, now with no degree values at all: X, A, Y along the top line; B and C below. Three fills are used, not two. The angle at B is red and so is ∠XAB; the angle at C is green and so is ∠YAC; and ∠BAC, the angle at A between the triangle's own sides, is filled brown, on a wider radius than the red and green wedges that flank it. So all three angles of the triangle turn up again along XY, each in its own colour, which is what makes the straight-angle step visible. The colouring is the proof made visible, and it is the single figure the explanation most needs to redraw faithfully.
The paper fold (Part I, §7.3, p.166). Checked against the printed page. Three stages are drawn: a whole paper triangle; the same with its top vertex folded down to the base, leaving a trapezium; and then the two base corners folded inwards, so the three angles sit side by side along the base edge.
The exterior angle figure (Part I, §7.3, p.167). Checked against the printed page. B, C and D lie along one straight line with D beyond C; A sits above. The angle between CA and CD is arced and captioned. Inputs: the figure and the construction of D as the extension of BC.
The exterior angle calculation (Part I, §7.3, p.167). Inputs: ∠A = 50°, ∠B = 60°. Two steps follow — the interior angle at C from the angle sum, and then the exterior angle from the straight line.
The relation left open (Part I, §7.3, p.167). The chapter asks the student to try other values of ∠A and ∠B, prints the two facts needed — the three interior angles total 180°, and the interior and exterior angle at C total 180° — and then stops with a question. It does not state the resulting relation. Checked against every page of the chapter. The explanation may derive it, and should say that it is completing the chapter's question rather than repeating the chapter's answer.

Figures to have open

Fig. 7.7 redrawn with the same three-colour coding: the angle at B and ∠XAB in one colour, the angle at C and ∠YAC in a second, ∠BAC in a third, so that the three colours end up side by side along XY. This is the proof in one picture.
A paper triangle that can be folded in three stages, ending with the three coloured corners abutting along the base. Redraw; do not lift the book's artwork.
A triangle with one side extendable, so the exterior angle opens up as the extension is drawn, and the interior and exterior angles at that vertex can be seen sharing an arm.
A straight line with three adjacent angles on it, usable on its own to establish that the three total 180° before the triangle is introduced.
No photograph, table or dataset from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 7 "A Tale of Three Intersecting Lines", §7.3 "Construction of Triangles When Some Sides and Angles are Given" — the third-angle question and Fig. 7.6 with the alternate-angle step, p.164; the arithmetic giving 60°, the Figure it Out set and the Try This triangle, p.165
Same part, within §7.3, the bold unnumbered block "Angle Sum Property", pp.165–166 — the general argument, Fig. 7.7, the naming of the property, the attribution to Euclid's The Elements, and the paper-folding check, p.166
Same part, within §7.3, the bold unnumbered block "Exterior Angles", p.167 — the definition and figure, the worked 50°/60° case, and the closing hint
Same part, SUMMARY, p.171, fifth bullet — the chapter's own one-line statement of the angle sum
Backward pointer: Part I, printed Chapter 5 "Parallel and Intersecting Lines", for alternate angles and for the straight angle