Which side is opposite and which is adjacent depends on the angle you pick

What to assume they know

• What a right triangle is, and that its two non-right angles are both acute
• That the hypotenuse is the side facing the right angle, and is the longest side
• Reading a labelled figure: vertex letters, the small square marking a right angle, and naming a side by its two endpoints
• Ratio of two lengths, and that a ratio of two lengths carries no unit
• The angle sum of a triangle, enough to see that the two acute angles of a right triangle add to 90°

What they should be able to do

• Identify the hypotenuse of a right triangle from the position of the right angle alone, without measuring
• Name, for a stated acute angle of a right triangle, which side is opposite to it and which is adjacent to it
• Re-name both legs correctly when attention moves to the other acute angle of the same triangle
• State what stays fixed under that change of viewpoint and what does not
• Explain why a side cannot be called "the opposite side" until an angle is named
• Recognise a right triangle inside a real situation — a tower and an observer, a balcony and a river bank, a balloon and the ground
• Say what the parts of the word trigonometry mean and what the subject relates to what

Where it usually goes wrong

• "BC is the opposite side, full stop." In Fig. 8.4 it is opposite to A; in Fig. 8.5, the same segment in the same picture, it is adjacent to C. The label belongs to the pairing of a side with an angle, not to the side.
• "Adjacent just means touching the angle." Two sides touch angle A — the leg AB and the hypotenuse AC. Adjacent is reserved for the leg, because the hypotenuse already has its own name and is never in the running.
• "The hypotenuse also changes when you switch angles." It cannot. It is pinned by the right angle, and the right angle does not move when you change which acute angle you are looking from.
• "Opposite means the longest side." The side opposite A is longer than the side adjacent to A only when A is the larger of the two acute angles. In the 24-7-25 triangle the leg opposite A is the short one.
• "You can call the sides opposite and adjacent as soon as you see a right triangle." Not until an acute angle is named. A figure with no angle chosen has a hypotenuse and two unlabelled legs.
• "These pictures are about towers and rivers." They are about a length you cannot measure sitting in the same triangle as an angle you can. The tower is interchangeable; the triangle is not.

Questions to check understanding

• Given a labelled right triangle and a named acute angle, list the hypotenuse, the opposite side and the adjacent side
• The same triangle asked again for the other acute angle, to test that the student re-derives rather than recalls
• Given a right triangle with two side lengths, find the third and then report the basic ratios at both acute angles
• True-or-false items on notation and naming, of the kind Exercise 8.1 question 11 poses — in particular the claim that one abbreviation stands for a different ratio's name
• Draw and label a right triangle to fit a described situation, marking which length is wanted and which angle is known

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• Fig. 8.1 (§8.1, p. 113). A drawing of the Qutub Minar with a dotted right triangle laid over it: the observer stands on the ground at one end of the horizontal leg, the tower rises as the vertical leg, and the sight line from observer to the top of the tower is the hypotenuse. The right-angle mark sits at the foot of the tower. The question the page attaches: can the height be found without climbing it? No numbers are printed on this figure.
• Fig. 8.2 (§8.1, p. 113). A house with a balcony on one bank, a small temple on the other, and dotted lines making a right triangle whose vertical leg is the height of the observer above the water and whose horizontal leg is the width of the river. Again no numbers are printed.
• Fig. 8.3 (§8.1, p. 114). Two balloons drawn at A and B above a common ground line, with two right-angle marks on that line and sight lines running back to two small figures at the left. This one carries two triangles in one picture, because the balloon has moved between the two sightings. No numbers.
• Fig. 8.4 (§8.2, p. 114) — the figure the whole topic turns on. Triangle ABC drawn with A at the lower left, B at the lower right, C above B, and the right-angle square at B. Three labels are printed inside the artwork: the sloping side AC is called the hypotenuse, the vertical side BC is called the side opposite to angle A, and the horizontal side AB is called the side adjacent to angle A. An arc marks the angle at A.
• Fig. 8.5 (§8.2, p. 115) — the identical drawing, re-labelled. The arc has moved to the angle at C. AC is still called the hypotenuse. BC now carries the adjacent label, this time with respect to C, while AB has become the leg facing C.
• The swap in numbers (Exercise 8.1 question 1, p. 121). A right triangle ABC with the right angle at B, AB = 24 cm and BC = 7 cm; the question asks for sine and cosine at A and then at C. Verified: the hypotenuse AC is 25 cm, since 24² + 7² = 576 + 49 = 625. Reading from A, the opposite leg is 7 and the adjacent leg is 24. Reading from C, the opposite leg is 24 and the adjacent leg is 7. So sine at A and cosine at C are the same number, 7/25, and cosine at A and sine at C are both 24/25 — the swap of the two legs shows up as a swap of the two answers.
• Word origins (§8.1, p. 114). The chapter breaks the name into three Greek pieces meaning three, sides and measure, and names Egypt and Babylon as the places where the oldest surviving work on the subject was set down, and reports that early astronomers reached for it to gauge how far away the stars and planets are.

Figures to have open

• Fig. 8.4 and Fig. 8.5 as a single shown step by step drawing with a switchable label layer. This is the chapter's own figure and the whole topic depends on the two states being visibly the same triangle. Redraw as a clean schematic rather than reproducing the printed art.
• A right triangle with the 24, 7 and 25 marked on the sides and both acute angles ringed, so the swap of section 9 can be read straight off it. Standard schematic.
• Simplified versions of Fig. 8.1, Fig. 8.2 and Fig. 8.3 — the tower, the balcony and the balloon — each with its right triangle drawn in and the right angle marked. Standard schematics; the printed originals carry no data the explanation needs.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 8 "Introduction to Trigonometry", §8.1 Introduction, pp. 113–114, including Fig. 8.1, Fig. 8.2 and Fig. 8.3
• §8.2 Trigonometric Ratios, pp. 114–115, up to the list of definitions, including Fig. 8.4 and Fig. 8.5
• Exercise 8.1 question 1, p. 121, supplies the numerical triangle used in section 9; Exercise 8.1 question 11 items (iii) and (iv), p. 121, are the naming traps
• The chapter summary, §8.5, p. 132, restates the three basic ratios in terms of the opposite, adjacent and hypotenuse labels introduced here