Inscribed Angle Theorem

वृत्तको कुनै पनि चापले परिधिमा बनाएको कोण (परिधि कोण), सोही चापले केन्द्रमा बनाएको कोण (केन्द्रीय कोण) को आधा हुन्छ भन्ने तथ्यलाई नै 'Inscribed Angle Theorem' भनिन्छ। गणितीय रूपमा भन्नुपर्दा, यदि एउटै चापमा आधारित केन्द्रीय कोण \(2\theta\) छ भने, त्यसको परिधि कोण सधैं \(\theta\) हुन्छ।

1. What is the center of the circle?
--Select-- Point \(A\) Point \(O\) Point \(C\)
2. If \(y\) is the central angle and \(x\) is the inscribed angle, what is the relation of \(x\) and \(y\)?
--Select-- \(x = y\) \(x = 2y\) \(y = 2x\)

Question

How do we learn geometry theorem?

Geometry learning is important because it serves formal logical thinking with spital reasoning. It supports to visualize axiomatic nature of mathematics to foster mathematical and cognitive development. As a learn one can use van Hiele model to learn geometry theorem because it postulates five levels of geometric thinking: visualization, analysis, abstraction, formal deduction and rigor. This hierarchical order helps to achieve better understanding. Initially, this model was focused solely on geometry, but it has now been extended to other areas of mathematics. According to the Van Hiele model, “inscribed angle theorem” can be learn as below”

Level 0: Visualization

Goal:Recognize shapes and configurations

1. See different figures and answer: “Which angles ‘sit on’ the circle? Which angle on the center?”
2. Learn “central angles” vs. “inscribed angles”
3. Use dynamic geometry software (e.g., GeoGebra) to drag points and observe the angles.
4. Language: “x is on the circle” and “y is on the center”

3. Which is the central angle?
--Select-- \(\angle ACB\) \(\angle AOB\) \(\angle OAC\)
4. Which is the inscribed angle?
--Select-- \(\angle ACB\) \(\angle AOB\) \(\angle OAC\)
5. Which arc made these central and inscribed angles?
--Select-- Arc \(\overparen{AB}\) Arc \(\overparen{AC}\) Arc \(\overparen{BC}\)
6. What is the name of the central angle? (Variable)
--Select-- x y o
7. What is the name of the inscribed angle? (Variable)
--Select-- x y o

Level 1: Analysis

Goal: Identify properties of geometric figures.
Measure at least three inscribed angles intercepting the same arc using protractors.

Complete the Table

Figure	Central angle (\(^\circ\))	Inscribed angle (\(^\circ\))
1
2
3

Notice pattern: “The inscribed angle is alwayshalf the central angle!”

Level 2: Abstraction / Informal Deduction

Goal: Understand relationships between properties; reason informally.
Categorize cases of the inscribed angle

1. Center on the angle’s side [construct an auxiliary radius to form an isosceles triangle]
2. Center inside the angle [decompose the angle into sums]
3. Center outside the angle [decompose the angle into difference]

Level 3: Deduction

Goal: Construct formal proofs.
Write proof for each case using two-column or paragraph format.

1. Case 1
2. Case 2
3. Case 3

Deductive proof of case 1

1. Given:Circle \(O\) where \(\angle ACB\) is inscribed by arc AB
2. To Prove:\(\angle ACB = \frac{1}{2} \angle AOB\)
3. Plan: auxiliary radius to form an isosceles triangle

SN	Statement	Reason
1	\(\triangle AOC\) is an isosceles triangle.	Definition
2	\(\angle OAC \cong \angle OCA=x\)	Base angles are congruent
3	\(y=\angle AOB = 2 \angle ACB=2x\)	Exterior Angle Theorem

1. In \(\triangle OCA\), if \(\angle OCA = x\), then what is the value of \(\angle OAC\)? (Given \(OA, OC\) are radii)
--Select-- \(x\) \(2x\) \(90 - x\)

Level 4: Rigor [Application & Extension]

Apply the theorem to prove

1. Thales’ Theorem: An angle inscribed in a semicircle is a right angle.
2. Equal Arcs → Equal Angles: If two inscribed angles intercept the same arc, they are congruent.
3. Cyclic Quadrilateral: Opposite angles sum to 180 degree
4. Reflect on Challenging Problem like “Can an inscribed angle ever be Reflex? Justify using the theorem.