Why every equilateral triangle has three 60° angles

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What to assume they know

Angles opposite equal sides are equal — angles facing equal sides are equal, and how that was proved
The three angles of a triangle add to 180° — Why the three angles of any triangle add to 180° (Part I, printed Chapter 7 "A Tale of Three Intersecting Lines", pp.165–166)
Equilateral means all three sides are the same length
Reading which angle faces which side in a lettered triangle
Dividing 180 by 3

What they should be able to do

State what equilateral asserts, and what it does not directly assert
Apply the equal-sides result to one pair of sides and name the angles it equates
Choose a second pair of equal sides and derive a second angle equality
Chain the two equalities to show all three angles are equal
Use the 180° total to obtain the single possible value
Explain why constructing and measuring is a check rather than a reason
Say which earlier result the whole deduction rests on
Identify congruent and equilateral triangles in built structures and in designs

Where it usually goes wrong

"Equilateral means all angles equal — that is the definition." It is not. The chapter defines the word by side lengths and spends a page and a half deriving the angle fact. If the angles were part of the definition there would be nothing to prove and the section would not exist.
"All angles 60° means all triangles with 60° angles are the same triangle." The three same-angle triangles on p.10 already showed that angles alone do not fix a triangle. Equilateral triangles come in every size; what they share is the 60°, not a size.
"You could have just measured one and seen 60°." Measuring one equilateral triangle tells you about that triangle and about the accuracy of your protractor. The chapter's closing sentence is explicit that the value was deduced.
"You need a different argument for each pair of sides." You need the same argument twice, applied to a different pair. Making that visible — the marks moving from one pair to another on an unchanged figure — is the whole teaching moment.
"∠A faces side AB." It does not; it faces BC. Getting the facing wrong is the most common way a student loses this derivation, and the two-figure sequence on p.18 is where to slow down and point.
"Three 60° angles proves the triangle equilateral." That is the converse. The chapter does not raise it. What the chapter shows runs one way: equal sides first, equal angles after.
"The photographs prove the mathematics." They are illustrations of where congruent triangles turn up, and the exercise attached to them asks the reader to describe what they see. They are not evidence for anything.

Questions to check understanding

State the size of each angle of an equilateral triangle and justify it
Given a triangle with all three sides marked equal, find the angles
Prove that an equilateral triangle has its three angles equal, naming the result used and saying how many times it is used
Given a figure with several equal-length segments marked, find the missing angles (Figure it Out question 6, Part II, p.21 — the chapter's largest such item)
Identify congruent triangles in a photograph or a design and describe the correspondence (the task attached to the five pictures, Part II, pp.19–20)
Explain why a construction confirms a result but does not establish it
The last bullet of the SUMMARY (Part II, p.22) is the chapter's own one-line form of this result

Examples worth working on the board

The first equilateral figure (Part II, §1.3, p.18). Checked against the printed page. ΔABC drawn with A at the apex, B at the lower left and C at the lower right. In this first copy a single tick sits on AB and a matching tick on AC — so the pair being used is AB and AC. Input: that pair. The equality ∠B = ∠C follows.
The second equilateral figure (Part II, §1.3, p.18). Checked against the printed page. The same triangle is reprinted immediately below, but the tick marks have moved: now one sits on AB and one on BC. Nothing about the triangle changed; only the pair being attended to did. This is the single most important visual beat of the topic, and it is easy to miss on the page because the two figures look almost identical. Show them as a before-and-after with the marks jumping.
The second equality (Part II, §1.3, p.19). From AB = BC the chapter gets ∠A = ∠C. Check the direction with the explanation: the angle facing AB is ∠C and the angle facing BC is ∠A, so those are the two that get equated. A script that guesses will get this backwards.
Chaining (Part II, §1.3, p.19). Inputs: ∠B = ∠C from the first pass and ∠A = ∠C from the second. All three follow.
The arithmetic (Part II, §1.3, p.19). Inputs: three equal angles, total 180°. The chapter writes the multiplication out as three times one angle equalling 180° before dividing.
The check (Part II, §1.3, p.19). The chapter then asks the reader to verify by construction. Note the ordering when explaining it: the deduction comes first and the drawing second. Reversing them would teach the opposite of the section.
Congruent triangles in real life (Part II, §1.3, pp.19–20). Checked against the printed page. Five captioned photographs: the glass pyramid at the Louvre in Paris, the Egyptian pyramid at Giza, a triangulated glass dome, a rangoli of coloured triangles arranged as a star, and Rabindra Setu — the Howrah Bridge — spanning the water. The reader is asked to describe the congruent triangles in each. Note for the script: the chapter's caption for these is about congruent triangles, not about equilateral ones, and only the rangoli and the Louvre pyramid faces read as anywhere near equilateral. Do not claim the photographs show equilateral triangles.
The SUMMARY (Part II, p.22). The last bullet is the 60° result in one line; the bullet above it is the isosceles result this topic used twice.

Figures to have open

An equilateral triangle whose tick marks can be moved between pairs of sides while the triangle itself stays put, with angle arcs that can light in pairs. This is the topic's central image and must be able to be shown moving.
A protractor overlay that can be laid on a 60° angle and read slightly off, for section 8.
Redrawn versions of the five real-world settings — a glass pyramid, a stone pyramid, a triangulated dome, a triangular rangoli, a steel truss bridge — with congruent triangles pickable out of each. The book's own photographs must not be reproduced; redraw or use freely licensed images.
No data table from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 1 "Geometric Twins", §1.3 "Angles of Isosceles and Equilateral Triangles", unnumbered subheading "Angles in an Equilateral Triangle", pp.18–19 — the definition by side lengths, the first tick-marked figure and the first equality (p.18), the second figure with the marks moved, the chaining, the arithmetic to 60° and the instruction to verify by construction (p.19)
Same part, same chapter, §1.3, "Congruent Triangles in Real Life", pp.19–20, with its five captioned photographs and the describe-what-you-see task
Same part, same chapter, §1.3, "Figure it Out", question 6, p.21
Same part, same chapter, SUMMARY, p.22, last two bullets
Backward pointer: Angles opposite equal sides are equal, the result this topic applies twice
Backward pointer: SAS, and why the angle has to be the included one for the p.10 figure showing that equal angles alone do not fix a triangle