SAS, and why the angle has to be the included one

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What to assume they know

SSS: three sidelengths fix a triangle completely — SSS, and the construction with two circles
Naming congruent figures so the correspondence is unambiguous — how to write a congruence so the pairing is clear
Drawing an angle of a given size at a given point with a protractor
Drawing an arc of a given radius from a given centre
The 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)
Knowing which angle sits between two named sides of a triangle

What they should be able to do

State the SAS condition and identify the included angle in a given triangle
Explain why knowing all three angles does not determine a triangle
Tell an included-angle set of givens apart from a non-included-angle one
Carry out the three-step construction for a base, an angle and a second side
Show, by construction, that the non-included case can produce two different triangles
Explain why the two SSA triangles are not related by a flip, and why the two SSS triangles were
State what SSA does and does not permit you to conclude
Given a labelled pair of triangles, decide whether SAS applies

Where it usually goes wrong

"Same angles means same triangle." The three triangles printed on p.10 have identical angles and different sizes. The chapter puts this case first precisely so that the sufficient conditions later are not mistaken for a list of everything that might work.
"Same angles means nothing at all." The other overcorrection. The three triangles do share something — the chapter says they have the same shape. What they fail to share is size, and size is half of congruence.
"SAS means any two sides and any angle." This is the misconception the whole second half of the topic exists to break. Say included every single time, and point at the corner while saying it.
"Two crossings always mean two answers." SSS also produced two crossings and was fine. The difference is where they sit: in the SSS construction the two points straddle the base and one triangle is the other flipped; in the SSA construction both points lie on the same ray, on the same side, and no flip carries one triangle to the other. This comparison is the heart of the explanation.
"SSA is a rule that is sometimes wrong." It is not a rule at all. It is a set of given information that fails to determine the triangle. Nothing is being computed incorrectly; there simply is not enough to go on.
"If the two triangles look different, at least one of them must be drawn wrong." Both ΔPQR and ΔPQS satisfy every given measurement exactly. Being drawn correctly is not the same as being determined.
"Because SSA fails, an angle that is not between the sides is useless." The chapter comes back to exactly that situation twice — in the right-angled case on pp.16–17 and in the AAS case on pp.14–16 — and both times it works. Failure in general is not failure always.

Questions to check understanding

Given three angle measures, explain why they do not determine a triangle
Identify the included angle for a stated pair of sides
Given two triangles with two sides and one angle marked equal, decide whether SAS applies or whether the case is SSA
Construct two non-congruent triangles from a stated SSA set of measurements
Given a labelled diagram, name the condition that establishes congruence and express the congruence (the shape of Figure it Out question 1, Part II, p.13)
Sort a list of measurement sets into those that guarantee congruence and those that do not (the shape of Figure it Out question 2, Part II, p.20 — its case (e) is exactly an SSA set)
The SAS bullet and the non-included-angle bullet of the SUMMARY (Part II, p.22) are the two statements the chapter expects back

Examples worth working on the board

The three-angle attempt (Part II, §1.2, pp.9–10). Inputs: 30°, 70° and 80°. Checked against p.10: three triangles are printed in a row, each lettered B at the lower left, C at the lower right and A at the apex, each carrying the same three angle marks — 30° at B, 70° at C, 80° at A — and each visibly larger than the one before.
The SAS case (Part II, §1.2, p.10). Inputs: AB and XY are both 6 cm; AC and XZ are both 5 cm; ∠A and ∠X are both 30°. Note that the equal angle is at A, the corner where AB and AC meet — that is what makes it the included angle. The chapter has the reader construct it and compare with classmates rather than printing a finished figure.
The SSA case (Part II, §1.2, p.10). Inputs: AB and XY are both 6 cm; AC and XZ are both 4 cm; ∠B and ∠Y are both 30°. The two sides named are AB and AC, and the equal angle is at B — one corner away. Only the second length has changed from 5 cm to 4 cm; the decisive change is which corner carries the angle. Make that explicit, because the two problems look almost identical on the page.
The rough diagram (Part II, §1.2, p.11). Checked against the printed page. A flat triangle with P at the lower left, Q at the lower right and R near the top; PQ is tagged 6 cm, RQ is tagged 4 cm, and the 30° arc sits at P. Note that P plays the part of B and Q the part of A in the problem statement.
The three construction steps (Part II, §1.2, p.11). Checked against the printed page. Step 1 draws PQ = 6 cm. Step 2 draws a ray l from P at 30° to PQ, printed with an arrowhead. Step 3 swings an arc of radius 4 cm, centred on Q, across l — and the printed step-3 figure already shows two crossings, R low down on the ray and S further out.
The two triangles (Part II, §1.2, p.12). Checked against the printed page. ΔPQR and ΔPQS are drawn side by side, both with the 6 cm base, both with the 30° at P, both with a 4 cm side to the third vertex. ΔPQR is squat and wide; ΔPQS is tall and leaning. They are the counterexample.
The distances along the ray. With PQ = 6 cm and ∠P = 30°, the perpendicular from Q to the ray is 3 cm, so an arc of radius 4 cm reaches the ray and crosses it at two points, both in front of P — roughly 2.5 cm and 7.8 cm along it. Those two numbers are added here, not the book's; they are here so the figure can place R and S correctly, and they should not be spoken as printed values.
Figure it Out, question 1 (Part II, §1.2, p.13). Checked against the printed page. Two triangles with A at the apex, B lower left, C lower right; and X at the apex, Y lower left, Z lower right. Labels: AB = 7 cm, BC = 5 cm, ∠B = 47° on the first; XZ = 7 cm, YZ = 5 cm, ∠Z = 47° on the second. This is a SAS item in which the equal angle sits at different-looking corners on the page, so the correspondence has to be worked out rather than read off left to right.

Figures to have open

Three triangles of increasing size carrying identical angle marks, for section 2. Standard schematic; keep the angle labels at the printed corners.
A single triangle in which the highlighted angle can be moved between corners, for section 7. This is the topic's key image and must be able to be shown moving.
The base–ray–arc construction, redrawn and able to be shown moving so the arc sweeps across the ray and both crossings appear in turn. Standard schematic.
The two finished SSA triangles side by side, both labelled with the same three measurements.
A side-by-side of the SSS construction and the SSA construction, for section 11. This comparison is not printed as a single figure in the book; it is assembled from the p.5 figure and the p.11 figure and should be built fresh.
No photograph or 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", §1.2 "Congruence of Triangles", pp.9–12 — the unnumbered subheading "Measuring the Angles" and the three same-angle triangles (pp.9–10), "Measuring Two Sides and the Included Angle" and the naming of SAS (p.10), "Measuring Two Sides and a Non-included Angle" with the rough diagram and the three construction steps (pp.10–11), and the two resulting triangles with the naming of SSA (p.12)
Same part, same chapter, §1.2, "Figure it Out", question 1, p.13
Same part, same chapter, §1.3, "Figure it Out", question 2, p.20, case (e)
Same part, same chapter, SUMMARY, p.22, the SAS bullet and the non-included angle bullet
Backward pointer: SSS: three sidelengths fix a triangle completely, whose two-crossings construction this topic deliberately re-runs with a different outcome
Forward pointer: RHS: the right-angled special case, the right-angled case where SSA works after all