# Cyclic quadrilateral

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# Additional Information

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## Reference Books

# Teaching Outlines

## Concept # 1. Cyclic quadrilateral and its properties

### Learning objectives

- A quadrilateral ABCD is called cyclic if all of its four vertices lie on a circle.
- In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
- If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.
- In a cyclic quadrilateral the exterior angle is equal to interior opposite angle.

### Notes for teachers

### Activity#1 Cyclic quadrilateral

- Estimated Time 10 minutes
- Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any

- Circle and quadrilaterals should have been introduced.

- Multimedia resources : Laptop
- Website interactives/ links/ / Geogebra Applets

- Process:

- The teacher can recall the concept of a circle, quadrilateral, circumcircle.
- Can explain a cyclic quadrilateral and show the geogebra applet.
- Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.

Developmental Questions:

- What two figures do you see in the figure ?
- Name the vertices of the quadrilateral.
- Where are all the 4 vertices situated ?
- Name the opposite interior angles of the quadrilateral.
- What do you observe about them.

- Evaluation:

- Compare the cyclic quadrilateral to circumcircle.

- Question Corner

- Can all quadrilaterals be cyclic ?
- What are the necessary conditions for a quadrilateral to be cyclic ?

### Activity No # 2.Properties of a Cyclic quadrilateral

- Estimated Time: 45 minutes
- Materials/ Resources needed

coloured paper, pair of scissors, sketch pen, carbon paper, geometry box

- Prerequisites/Instructions, if any

- In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
- In a cyclic quadrilateral the exterior angle is equal to interior opposite angle

- Multimedia resources
- Website interactives/ links/ / Geogebra Applets

This activity has been taken from the website http://mykhmsmathclass.blogspot.in/2007/11/class-ix-activity-16.html

- Process:

- Draw a circle of any radius on a coloured paper and cut it.
- Paste the circle cut out on a rectangular sheet of paper.
- By paper folding get chords AB, BC, CD and DA in order.
- Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained.
- Produce AB to form a ray AE such that exterior angle CBE is formed.
- Make a replica of cyclic quadrilateral ABCD using carbon paper.
- Cut the replica into 4 parts such that each part contains one angle .
- Draw a straight line on a paper.
- Place the two opposite angles, angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
- Place other two opposite angles, angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
- Make a replica of angle ADC and place it on angle CBE . Write the observation.

Developmental Questions:

- How do you take radius ?
- What is the circumference ?
- What is a chord ?
- What is a quadrilateral ?
- Where are all four vertices of a quadrilateral located ?
- What part are we trying to cut and compare ?
- What can you infer ?

- Evaluation:

- Angle BAD and angle BCD, when placed adjacent to each other on a straight line, completely cover the straight angle.What does this mean ?
- Angle ABC and angle ADC, when placed adjacent to each other on a straight line, completely cover the straight angle.What can you conclude ?
- Compare angle ADC with angle CBE.

- Question Corner:

Name the two properties of cyclic quarilaterals.

## Concept # 2.Construction of cyclic quadrilateral

### Learning objectives

- Ability to construct a cyclic quadrilateral accurately .

### Notes for teachers

### Activity No # Constructing a cyclic quadrilateral

- Estimated Time: 40 minutes.
- Materials/ Resources needed:

- Laptop, geogebra file, projector and a pointer.
- Students constructing materials, the geometry box.
- white papers.

- Prerequisites/Instructions, if any

- Sufficient knowledge regarding construction of perpendicular lines, bisectors, angles and circle.

- Multimedia resources : Laptop
- Website interactives/ links/ / Geogebra Applets: For step by step illustration of cyclic quadrilateral construction please refer to the website: http://www.matrusrieppower.net/Constructionoftriangleandcyclicquadrilateral.html.
- Process:

- The teacher can do this activity after introducing the concept and properties of cyclic quadrilateral.
- She can project the file and let students watch it carefully.
- After watching discuss the steps of construction and the purpose of each step so that the students can appreciate the sequence of construction steps.
- Then ask the students to actually construct a cyclic quadrilateral for the given measures.

- Developmental Questions:

- What is a cyclic quadrilateral ? Why is it called so ?
- Name the measuring parameters of it ?
- What measures are given for its construction ?
- Explain the steps involved in determing the radius of the required circle ?
- What do the measures of the arcs specify ?

- Evaluation:

- Were the students able to justify the sequence of steps involved ?

- Question Corner:

- Can you draw a circle first and then the quadrilateral ? Why not so ?

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

## Concept # 3. Theorems on cyclic quadrilaterals

### Learning objectives

- Both pairs of opposite angles of a cyclic quadrilateral are supplementary.
- When one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle.

Converse theorems:

- Suppose a quadrilateral is such that the sum of two opposite angles is a straight angle, them the quadrilateral is cyclic.
- If the exterior angle of a quadrilateral is equal to the interior opposite angle, then the quadrilateral is cyclic.

### Notes for teachers

### Activity No 1. Theorems

- Estimated Time : 40 minutes.
- Materials/ Resources needed:

Laptop, geogebra file, projector and a pointer.

- Prerequisites/Instructions, if any

- A cyclic quadrilateral and its properties.
- The linear pair and exterior angle theorem.
- The circle theorem (Angle at centre = double the angle at the circumference)

- Multimedia resources: Laptop
- Website interactives/ links/ / Geogebra Applets:

This geogebra file was done by ITfC-Edu-Team.

- Process:

- The teacher can project the geogebra file and prove the theorems.

- Developmental Questions:

- How many angles does a cyclic quadrilateral have ?
- Name the opposite angles of it.
- Name the minor arc.
- Recall the angle -arc theorem.
- What is the total angle at the centre of a circle ?
- Name the angles at the centre of the circle.
- What is the sum of those two angles ?
- How can you show that <b and <d are supplementary from above observations ?

- Evaluation;

- What is the converse of this theorem.

- Question Corner;

- Write down the steps to prove the converse of this theorem.

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

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