Copying an angle, and why triangle congruence proves it works

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What to assume they know

The perpendicular bisector, and the equidistance property that justifies it — arcs, radii, and reading equalities off congruent triangles
The SSS condition for triangle congruence (SSS: three sidelengths fix a triangle completely)
Transversals, and the corresponding angles a transversal makes (A transversal creates two matching sets of four angles and Corresponding angles are equal exactly when the lines are parallel; Part I, printed Chapter 5 "Parallel and Intersecting Lines", §5.5–§5.6)
The earlier ruler-and-set-square way of drawing a parallel line (Drawing a parallel line using the corresponding-angle test; Part I, printed Chapter 5, §5.7)
Recognising an isosceles triangle from two equal sides

What they should be able to do

Explain why a compass alone cannot transfer an angle directly
Describe how cutting both arms at one radius turns an angle into a triangle
Identify the one length that has to be carried across, and say why the other two are free
Carry out the five-step angle-copying construction
State the congruence condition that proves the copy is exact
Explain how copying a corresponding angle produces a parallel line
Carry out the parallel-line construction with a compass and an unmarked ruler
Use repeated copies of one unit, in two orientations, to build a repeating band

Where it usually goes wrong

"Just measure the angle and draw it again." That is the protractor answer, and it is the one the chapter has ruled out since p.140. It is also less exact: a protractor reading is rounded, a carried compass span is not.
"The compass copies the angle." It copies one length. The angle is reconstructed by congruence afterwards. Keep those two steps visibly separate in the figure.
"The arc radius has to be some particular size." It does not — but it has to be the same at A and at X. Every equality in the proof comes from that single reuse.
"You have to carry all three sides across." Two of them are already equal because one radius made them. That is the economy the isosceles remark on p.146 is pointing at.
"Parallel means the same distance apart, so this construction is about distance." The construction never measures a gap. It works through the equal corresponding angles the chapter recalls on p.147.
"Two orientations means two different units." Fig. 6.6 repeats one shape; what alternates is how it is turned. If the explanation draws two shapes, it has broken the argument of the section.

Questions to check understanding

Copy a given angle using a compass and an unmarked ruler, and name the congruence condition used
Copy several angles drawn in different orientations without measuring anything (Figure it Out question 1, Part II, p.147)
Reproduce the repeating band (Figure it Out question 2, Part II, p.147)
Given a line and a point off it, construct the line through that point parallel to the given line
Construct four parallel pairs, each set at a different slant (Figure it Out question 1, Part II, p.148)
Reproduce the eight-pointed star figure (Figure it Out question 2, Part II, p.148)
The bisected-and-copied bullet of the SUMMARY (Part II, p.163) is the statement the chapter expects back

Examples worth working on the board

Fig. 6.6 (Part II, §6.1, p.145). Checked against the printed page. A horizontal band of five congruent circular sectors in strict alternation — three of them with the apex down and hatched with grey parallel lines, and between those, two with the apex up and left white — so the outline rises and falls in a wave. Only three of the five carry the hatching. Directly under it the chapter prints the single unit twice on its own, once curving one way and once the other, which is what "two orientations" means here.
The five printed steps (Part II, §6.1, pp.146–147). Checked against the printed page. Step 1: the given angle with its corner lettered A, beside a bare ray with its end lettered X. Step 2: an arc struck from A cutting the two arms at B and at C. Step 3: an equal-radius arc struck from X, cutting the ray at Z. Step 4: the compass span BC carried over and marked off from Z along that arc, landing at Y. Step 5: the two figures side by side, each carrying its arms, its arc and its three lettered points; the chapter never rules in BC or YZ, so the two triangles are implied rather than drawn.
The three equalities (Part II, §6.1, pp.146–147). Inputs: AB = XY and AC = XZ, both because one radius was used four times; and BC = YZ, because that span was carried across. Those three are what SSS wants.
Fig. 6.7 (Part II, §6.1, p.147). Checked against the printed page. A horizontal line m with a slanting line l crossing it at a point lettered A, and the angle between them tagged with a small italic a. In the figure just below it, a point B is marked further up l, a second horizontal line runs through B, and the angle at B is tagged with the same italic a.
Figs. 6.8–6.10 (Part II, §6.1, p.148). Checked against the printed page. In Fig. 6.8, one radius is used from A and from B; the arc from A cuts m at C and cuts l at D, and the arc from B crosses l at the point Fig. 6.9 letters F — in Fig. 6.8 that crossing is drawn but left unlettered, so anyone drawing it must not show the F a figure early. In Fig. 6.9 the span CD is carried to the arc at F, landing at E. In Fig. 6.10 the line through B and E is drawn and marked parallel to m, and lettered n. Note the letter order printed in the captions and do not renumber it.
The eight-pointed star (Part II, §6.1, "Figure it Out", question 2, p.148). Checked against the printed page. Eight narrow grey kites radiating from one centre with eight white ones between them; the outer points are lettered S, T, U, V, W, X, Y, Z and the inner ring A, B, C, D, E, F, G, H. The figure is set as a parallel-lines exercise.

Figures to have open

The repeating band and its single unit, able to be shown moving so one unit can be lifted out, flipped, and set back down. Redraw rather than reproduce.
An angle-copying rig: a source angle on the left, a bare ray on the right, and a compass that visibly keeps its span between the two. This is the topic's key image.
The two triangles ABC and XYZ pulled out side by side with their three equal sides marked. The chapter leaves BC and YZ unruled, so both triangles must be built fresh.
The transversal figure with the two corresponding angles tagged, able to be shown moving so the second angle can be built by copying the first.
The eight-pointed star of p.148, with the parallel pairs highlightable.
No photograph, table or dataset from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 6 "Constructions and Tilings", §6.1 "Geometric Constructions", the unnumbered subheading "Repeating Units and Repeating Angles", p.145, with Fig. 6.6
Same part, same chapter, §6.1, "Steps of Construction to Copy an Angle", pp.146–147, ending with the SSS conclusion at the top of p.147
Same part, same chapter, §6.1, "Figure it Out", questions 1–2, p.147
Same part, same chapter, §6.1, "Construction of a Line Parallel to the Given Line", p.147, with Fig. 6.7, and p.148 with Figs. 6.8–6.10
Same part, same chapter, §6.1, "Figure it Out", questions 1–2, p.148
Same part, same chapter, SUMMARY, p.163, the bisecting-and-copying bullet
Backward pointers: The perpendicular bisector, and the equidistance property that justifies it, Bisecting an angle, and halving 90° to get 45°
Forward pointer: Constructing 60° from an equilateral triangle, and the arches built on it, where copied equal angles set up the arch supports