Off the circle the angle changes: points inside and outside compared (Fig. 5.25)

What to assume they know

• Equal angles in the same segment: the arc looks the same from every point beyond it — that all points of a circle off a given arc see it at one angle
• Major and minor arcs, and why an arc's central angle is double what it subtends on the circle (Theorem 9) — the doubling relation
• The exterior-angle theorem, and that an exterior angle strictly exceeds either remote interior angle
• Chord, and telling an interior point from an exterior point of a circle
• That the angle at which a segment is seen can be compared between two positions without either being measured

What they should be able to do

• State that the equal-angle property holds only for points of the circle, and say what would follow if it held everywhere
• Sort the lettered points of Fig. 5.25 into inside, on, and outside, and say which group gives a single reading
• Explain, using an exterior angle, why an interior point gives a larger angle than an on-circle point and an exterior point a smaller one
• Say where in the chapter that direction is established, and that Fig. 5.25's own paragraph does not supply it
• Given a chord, an on-circle reading and a measured angle at an unknown point, decide whether the point is inside, on, or outside the circle
• Explain why this negative result is what makes the concyclicity test in §5.8 informative
• Distinguish "the angle differs" from "the angle differs in a stated direction", and say which the chapter asserts where

Where it usually goes wrong

• "A fixed chord looks the same from everywhere." It does not; that is the content of this topic. Everyday intuition about apparent size says the opposite of what students often assume here — they expect variation off the circle and constancy is the surprise, or they expect constancy everywhere and the variation is the surprise. Fig. 5.25 settles it either way.
• "The chapter says inside gives a bigger angle." On p. 110 it says only that the readings differ. The direction comes from p. 112. An explanation that attributes the direction to Fig. 5.25's paragraph is putting words in the book's mouth.
• "All the interior points give one value and all the exterior points give another." No — within each region the values vary from point to point. That is why the chapter names three interior points and two exterior ones rather than one of each.
• "Points on the circle but on the other arc count as 'on the circle', so they agree." They give the supplementary value. The equality is for one side of the chord.
• "If the angle is different the point could be anywhere." Given the side, the angle pins the region: bigger than the on-circle reading means inside, smaller means outside. That is a decision procedure, and it is what Theorem 10 runs.
• "This is a side remark before the real section." It is the load-bearing observation. Without it §5.8's test would be vacuous, because a test that every point passes distinguishes nothing.

Questions to check understanding

• Given a chord, an on-circle reading, and a measured angle at an unmarked point on the same side, say whether that point is inside, on, or outside the circle
• Explain why an angle taken at an interior point exceeds the angle taken at a point of the circle on the same side
• Given a lettered figure, sort the points into inside, on, and outside, and say which angles must be equal
• Exercise Set 5.6 Q2(i) and Q2(iii), p. 111 — and for Q2(iii), state the same-side condition the printed wording omits before answering it
• Explain why the "same side" condition cannot be dropped from a statement about equal angles
• Explain why the equal-angle property being false off the circle is necessary for the concyclicity test to work
• One-mark: two points on a circle on opposite sides of a chord — are their angles equal?

Examples worth working on the board

The chapter prints no answers and no measurements here, so anything marked verified or measured by me is added here, not the book's.

• Fig. 5.25 (p. 110; the caption names points and the chord). This is the whole topic and it must be read off the page, not the text layer. A circle with A high on the left and B high on the right, both on the circle, and the chord AB drawn between them. Seven further lettered points, each joined by a segment to A and to B, so the figure is a fan of angles all standing on AB:
  • inside the circle: I, just below and right of A; C, near the middle; H, right of centre and lower;
  • on the circle: D, on the lower left arc; E, on the lower right arc;
  • outside the circle: G, just below the lower arc; F, well below and left, outside. The printed paragraph beside the figure states the classification — F and G outside, I, C and H inside, D and E giving equal angles — so the grouping is the chapter's, not an inference from the drawing.
• What the chapter asserts, precisely. Its paragraph says the readings at F and G differ from each other, that the three interior readings all differ, and that every point of arc AB gives the same reading, so the readings at D and E are equal. It does not say which of inside or outside gives the larger value. Getting that distinction into the explanation is the point of section 6 — the chapter is being careful, and reporting it as though it had claimed more would be a misreading.
• Where the direction is established (§5.8, p. 112, inside Theorem 10's argument). With E on the circle and D off it: if D is outside, the angle AEB is exterior to triangle BED, so the on-circle reading exceeds the outside reading. If D is inside, the angle ADB is exterior to that triangle instead, so the inside reading exceeds the on-circle reading. Verified, stacking the two: interior point > on-circle point > exterior point, for a fixed chord and a fixed side. Cite p. 112 for it, not p. 110.
• A consistent numerical set the explanation can build (not in the book; the chapter prints no numbers in this figure). Chord AB, with every point of the far arc reading 50°. Verified: an interior point reads more than 50° — say 68°; an exterior point reads less — say 34°. Then the exterior-angle relation shows itself: the 50° at the on-circle crossing point plus the remaining angle of the small triangle account for the 68°, and symmetrically for the 34°. Keep whatever numbers are chosen consistent with an actual drawing, because a student will check.
• The locating test, worked. Inputs: a chord AB, an on-circle reading of 50° for one side, and an unknown point P on that side at which the angle APB measures 61°. Verified: P is inside the circle. At 50° it would be on the circle; at 41° it would be outside. This is the reasoning of Theorem 10 turned into something a student can do, and it is the cleanest way to make section 8 concrete.
• Exercise Set 5.6, Q2(iii) (p. 111). Inputs: X and Y are not on the circle, and the angles AXB and AYB are equal; is Y picked up by the circle drawn through A, B and X? Verified: yes once the same-side condition is put back, which the printed wording omits — and the reason is this topic's: equal angles from the same side force the same region, and the circle through A, B and X is then the circle both must lie on. Drop the condition and the answer is no. Put X and Y symmetrically on opposite sides of AB, each seeing it at 120°, and the circle through A, B and X has Y as its centre, so it never reaches Y; only the 90° case survives on opposite sides. Note this is the same point the "same side" caution below makes, so the two must not disagree. The chapter is placing the question immediately before Theorem 10, which proves it — with that hypothesis stated.
• The "same side" caution (for section 10). Inputs: a chord AB, a point on the major arc, a point on the minor arc. Verified: the two readings are supplementary, not equal — 50° and 130°. So "equal angles" cannot be used to conclude anything without the same-side condition, and Theorem 10 on p. 111 states that condition explicitly. Two points giving 50° and 130° are both on the circle and the naive test would reject them.

Figures to have open

• Fig. 5.25 redrawn, and redrawn with the three regions visually distinguished. The printed figure (p. 110) is a dense fan of chords in one weight and it is genuinely hard to see at a glance which points are inside, on, and outside — which is the one thing the figure exists to show. Colour or line-weight coding is the fix.
• A single small triangle isolating the exterior-angle step: the chord, an on-circle point, an off-circle point on the same ray, and the exterior angle marked. Standard schematic; the chapter's version of this argument is prose on p. 112 with Figs. 5.27 A and B beside it.
• A sliding-point movement for section 8: one point moving along a ray from inside to outside, its angle reading shown falling and passing through the on-circle value exactly once. Standard schematic; not in the book, and it makes the locating test obvious.
• A supplementary-pair figure for section 10: one chord, two on-circle points on opposite arcs, readings marked. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 5, "I’m Up and Down, and Round and Round", §5.7.1. Fig. 5.25 with its caption, and the paragraph beside it classifying the points, are on p. 110.
• The sentence naming this as the property that distinguishes the circle from other shapes is at the top of p. 110, immediately above the Fig. 5.25 paragraph.
• The direction of the two inequalities is established on p. 112, inside Theorem 10's argument, with Figs. 5.27 A and B; see Equal angles on the same side force concyclicity (Theorem 10).
• Exercise Set 5.6, Q2, all three parts on p. 111. Theorem 10 as stated, p. 111.
• Chapter Summary, p. 117 — the concyclicity bullet, which states the test this topic makes informative.