Angles in Circles: 6.0

• Indicator of Progress
• Teaching Strategies

Supporting materials

• Related Progression Points
• Developmental Overview of Space (PDF - 32Kb)
• Developmental Overview of Working Mathematically (PDF - 31Kb)

 

Indicator of Progress

Success depends on students extending their ability to follow formal mathematical arguments and to use deductive reasoning to prove or disprove conjectures. At this level students are able make conjectures based on their observations of angles in circles associated with chords and tangents and to reason deductively from known properties of triangles to explain why certain conjectures about angles are true.

This work builds on triangle geometry, in particular. It also extends students’ understanding of mathematical discovery and proof. They know that a result cannot be proved by just a few examples. They are able to follow longer arguments than before, dealing with more complex content.

 

Illustration 1: Confusing data with proof

At this level, students should understand that showing a relationship holds in some cases, even in very many cases, is not sufficient to prove it for all cases. Instead, logical argument deducing the relationship from a firm basis of known facts is required.

 

Illustration 2: Finding special cases of a general relationship

Students may see certain properties as separate unrelated properties rather than being able to recognise the property as a special case of a more general relationship.

 

Illustration 3: Recognising figures within complex diagrams

Students may have difficulty recognising known geometric figures (for example, triangles that could be congruent, angles standing on the same arc) within a diagram. They are therefore unable to find a starting point from which to prove relationships in the diagram.

Students’ experience should not be confined always to conventional arrangements: show them the variation.

 

Teaching Strategies

It is important that students do not simply memorise geometric relationships. Instead, the emphasis should be on reasoning. Geometric reasoning may be associated with a variety of activities:

• Students should be given the opportunity to explore angle relationships through the use of interactive geometry software (also called dynamic geometry), such as The Geometer’s Sketchpad, Cabri Geometry or the free downloadable software GeoGebra (http://www.geogebra.com). This exploration should encourage the formulation of conjectures, which students should then try to prove by reasoning from their existing knowledge of geometric properties.
• They should be able to apply their existing knowledge of properties to explain other properties.
• They should be able to engage in reasoning to determine unknown angles in diagrams incorporating angles in circles.

Interactive geometry can be used in several ways:

• students construct and use their own files (requires computers and construction skill),
• students use files prepared by the teacher (requires computers),
• teacher demonstrates the files to students, who pool measurements (requires one computer and data projector).

If interactive geometry is not available, then students can each construct a different case of a general property by-hand, and pool the results of measurements to provide empirical data for conjecturing.

Activity 1: Angle at the centre discusses a common teaching pattern for establishing formal arguments, beginning with experimentation, leading to conjecture, and then to a logical argument.
Activity 2: Angel in a semicircle provides an example of a proposition that is proved when it is seen to be a special case of a previously known proposition. Interactive geometry is valuable to explore special cases.
Activity 3: Tangents from a point works only from a deductive approach, where students work from what is known, using a construction line, to derive a formal argument.
Activity 4: Angle chasing gives an example of a type of problem that provides many opportunities to strengthen powers of deduction and to build fluency in recalling geometric facts.

 

Activity 1: Angle at the centre

The goal of this activity is for students to know that the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. In addition, they should know why this result is true.

The plan is in three stages:

• Using interactive geometry (or sharing data from several by-hand constructions), students collect and tabulate data
• They conjecture what the relationship might be
• With scaffolding from the teacher, they produce a formal argument establishing the truth of the conjecture, based on their knowledge of properties of isosceles triangles and exterior angles of triangles.

Step 1: Generating and tabulating data

Students use an interactive geometry file to generate and tabulate empirical data on the angle at the centre and the angle at the circumference. The screen dump below is from GeoGebra.

Angle at centre	Angle at circumference
122.89°	61.45°
121.18°	60.59°
114.96°	57.48°
66.14°	33.07°

 

 

 

 

 

Step 2: Conjecturing

Students should notice that the angle at the centre is always twice the angle at the circumference (sometimes there is a discrepancy in the second decimal place due to rounding). It is important that students recognise that the apparent relationship they have observed has not been proved by the data. However, the evidence from dragging points A, B and C strongly supports the conjecture. The next stage is to prove why this relationship exists.

Step 3: Proving

Given time in the exploratory environment of interactive geometry software, some students may suggest joining C and O to make two triangles. (See Diagram 1 below). Adding a construction line is an important strategy that students should learn.

It may be useful at this stage for students to make a diagram on paper, so they can mark all the information that they have on the diagrams easily.

They should also recognise that the two triangles OAC and OBC are isosceles triangles because in each case two of the sides are radii of the circle. Extending the segment CO then creates exterior angles for the two triangles (ÐAOD and ÐBOD).

At this point, students can use the property that the exterior angle of a triangle (in this case k) is equal to the sum of the 2 of the interior angles (in this case j+j = 2j ). Hence k = 2j.

Similarly, n = 2m. (See Diagram 2). Hence k + n = 2j + 2m =2(j + m), as we wanted to prove.

Alternatively, without knowing about the exterior angle property, students can reason that ÐAOC =180 - 2j and ÐDOA= k =180 - (180 - 2j ), so k= 2j. Similarly, n = 2m.

Students should understand the generality incorporated in this proof: that the relationship holds for all arcs AB on the circle and for all points C on the arc.

 

 

Activity 2: Angle in a semicircle

After students have met a relationship, then exploring special cases is a useful way of finding out new facts, as well as reviewing what has been learned.

Exploring special cases is also a useful strategy to see the variation in possibilities for a relationship, and to familiarise students with the different ways in which a given relationship may appear.

Using interactive geometry, it is easy to look at special cases. In the image below, the angle at the centre (O) has been systematically varied by moving point A from zero through position K, to a standard view at position L, to a straight angle (position M) and beyond (position N).

These diagrams extend our view of the angle at the centre theorem discussed in Activity 1, and the interactive geometry shows empirically that the relationship continues to hold (until point A reaches C).

If interactive geometry is not available, then different students in a group can each sketch a different case, and measure their own examples, pooling the results and showing them to each other. It is important in the unusual cases (e.g. cases K and N below) that the angle at the centre and the angle at the circumference are clearly identified by students.

Using the same interactive geometry construction as for Activity 1, students can drag point A or B so that AOB is a straight line, that is, AB is a diameter. They should notice that ÐACB is then a right angle, and recognise that this is a special case of the proof in activity 1: when the angle at the centre is 180 o, then the angle at the circumference is 90 o.

Some students may like to prove the property afresh, without reference to the argument in Activity 1. The construction line OC produces two isosceles triangles.

 

Activity 3: Tangents from a point

At Level 6.0, students can follow formal mathematical arguments for the truth of propositions. However, this should not only be a presentation by the teacher, which students follow. Instead, students should contribute to developing the argument if possible, and the teacher should share some of the process of proof discovery with them.

Diagram 1 shows two tangents (PA and PB) to a circle from point P. The goal is to show that PA = PB and show that Ð APO = ÐBPO. (Assume students know that tangents are perpendicular to the radius).

An important problem solving strategy is to be clear what you want to find (in this case that 2 lines and 2 angles are equal), and what you know already. In diagram 1, it is known that the two radii of the circle OA and OB have equal length. It is also known that the angles ÐOAP and ÐOBP are right angles because AP and BP are tangents. Mark this information on the diagram.

At this point it is not clear what to do next. Students may suggest using kite properties, but this approach does not seem fruitful. Since more is known about triangles than quadrilaterals, it is likely to be useful to construct the line OP to make two triangles. Making a construction is often, as in this case, the key to a successful proof.

By drawing segment OP, students should recognise that they need to show that triangles AOP and BOP are congruent. They should also recognise that the side OP is common to both triangles, and mark this on the diagram, resulting in Diagram 2. At this stage, triangles AOP and BOP are known to be right angled triangles with two pairs of sides equal. This should lead at least some students to make a connection with Pythagoras’ theorem to show that the third side of the two triangles must also be equal, and therefore that the triangles are congruent. The equality of the sides and angles then follows.

After seeing this argument or constructing it themselves, students should write out each step carefully, so that the truth of the new proposition is clearly shown to follow from previously known facts.

 

Activity 4: Angle chasing

Angle chasing exercises, starting with a few known angles or sides and deriving many others, can be rewarding as puzzles, as well as providing many opportunities to strengthen powers of deduction and to build fluency in recalling geometric facts.

The following diagram is one of many which incorporate angles standing on the same arc, and tangents. The instruction to find the size of as many angles as you can (rather than a specific angle) has interesting pedagogy. Some research suggests that open tasks such as this enable students to recall all the relevant facts that they know, and establish them better in students’ memory than a task that focuses their attention on just one goal.

Angle BOC = 130 o and EC is a tangent to the smaller circle at point C.

Find the sizes of as many angles as you can and show that BC is parallel to EF.