# Angles

Philosophy of Mathematics |

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## Contents

- 1 Concept Plan
- 2 Textbook
- 3 Additional Information
- 4 Teaching Outlines
- 4.1 Concept #1.What is an angle ?
- 4.2 Activities
- 4.3 Concept #2. Using a Protractor- Measuring an angle
- 4.4 Concept #3.Types of angles
- 4.5 Concept # 4. Angle constructions
- 4.6 Concept # 5. Angle bisector-Its construction
- 4.7 Concept # 6. Pairs of angles
- 4.8 Concept # 7.Angles formed when lines are cut by a transversal

- 5 Hints for difficult problems
- 6 Project Ideas
- 7 Math Fun

### Concept Plan

# Textbook

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# Additional Information

## Useful websites

## Reference Books

# Teaching Outlines

#### Concept #1.What is an angle ?

### Activities

Foramtion of

## Concept #2. Using a Protractor- Measuring an angle

### Learning objectives

### Notes for teachers

### Activity No #

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

### 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.Types of angles

### Learning objectives

### Notes for teachers

### Activity No #1.Crazy Angles using Geogebra

- Estimated Time: 40 minutes
- Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any:

- The students should have a basic understanding about point, rays, line segments and vertex.
- They should know how angles are formed.
- They should know that angles are measured in units called degrees. 360 ° is a full rotation (a circle)
- They should know to use a protractor and measure the angles.
- They should know the meaning of terms acute, obtuse, straight, reflex, and complete angles.

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

- Process:

- The teacher should recaptulate the concept of a point, line segment, ray, vertex and angles.
- The teacher should show how angles are formed.
- Discuss the concept of cartesian plane, X and Y axes, rotation, and how it relates to angles.
- Demonstrate how to measure angles using a protractor.
- Define and illustrate the classification of the types of angles—acute, obtuse, right, straight zero and complete angles.
- In the succeeding class give the students protractors and let them have enough practise measuring and classifying angles.

Developmental Questions:

- What is a point ?
- A minimum of how many points are needed to define a line segment ?
- A minimum of how many points are needed to form an angle ?
- Name the line segments from the figure.
- What is a vertex ?
- How many rays /line segments are needed to form an angle ?
- Name the vertex at which the angle is formed
- Name the angle .
- Name the type of angle formed.

- Evaluation:

- Assess the students knowledge of angles by projecting different types of angles and asking them to name
- What are the characteristics of an acute angle ?
- What are the characteristics of an obtuse angle?
- What are the characteristics of a right angle
- Evaluate if the students have understood that :

- An angle is formed where 2 lines meet at a point.
- A right angle looks like a corner of a square or a rectangle.
- An acute angle is narrower than a right angle.
- An obtuse angle is wider than a right angle.
- Question Corner:

- What is an angle ?
- Where do you name an angle ?
- How do you identify different types of angles in 2-dimensional figures?
- How do angles help to classify 2-dimensional figures?
- Are angles <ABA' and <A'BA the same ? Justify
- Differentiate between the zero angle and a complete angle.

## Concept # 4. Angle constructions

### Learning objectives

### Notes for teachers

### Activity No #

## Concept # 5. Angle bisector-Its construction

### Learning objectives

### Notes for teachers

Activity - Construction of angle with measure 22.5∘

This activity helps to illustrate the 'angle bisector' construction three times since we construct ∡22.5 by constructing ∡90∘ (bisecting a segment / straight angle ∡180∘, then bisect ∡90∘ to get ∡45∘ and finally bisect ∡45∘ to get ∡22.5∘).

## Concept # 6. Pairs of angles

### Learning objectives

### Notes for teachers

### Activity No #

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

### Activity No #

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

### 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 # 7.Angles formed when lines are cut by a transversal

### Learning objectives

### Notes for teachers

**Activity No # 1.Angles formed when a transversal intersects parallel lines **

***Estimated Time :** 40 minutes

***Materials/ Resources needed :**Laptop, geogebra file, projector and pointer.

***Prerequisites/Instructions, if any :**

- The students should have prior knowledge of parallel lines , transversal, angles and types of angles formed when a pair of parallel lines are intersected by a transversal.
- They should know what the terms interior, exterior, adjacent, alternate, consecutive, congruent, linear and corresponding mean.
- Students should know the definition of complementary angles, supplementary angles, and congruent angles.

***Multimedia resources:**
Laptop

***Website interactives/ links/ / Geogebra Applets**

- This is a resource file on 'vertically opposite angles'

It has been created by Sucheta, Mathematics teacher, GHS Thyamangondlu

***Process:**

- Reiterate that when a transversal intersects parallel lines, several pairs of congruent and supplementary angles are formed.
- Have students draw two parallel lines and a third line(transversal) intersecting those two lines on their own paper. Direct them to think about any angle relationships they see. Have them discuss their conjectures with a partner.
- The teacher can next project the GeoGebra worksheet and discuss about types of angles and their relationships with the class .
- Finally the teacher and students can summarize together the angle relationshipsalong with their characteristics.

Linear pair of angles - adjacent and supplementary

- Vertical angles - congruent
- Corresponding angles -congruent
- Alternate interior angles - congruent
- Same side interior angles - supplementary
- Alternate exterior angles - congruent
- Same side exterior angles - supplementary

***Developmental Questions :(What discussion questions)**

- How many pairs of corresponding angles are there ?
- What is true about corresponding angles formed when parallel lines are cut by a transversal?
- Compare different pairs of alternate interior angles. What do you notice?
- <EGD and <AHF are alternate exterior angles. What is another pair of alternate exterior angles?
- Compare different pairs of same-side interior angles. What do you notice?
- Compare different pairs of same-side exterior angles. What do you notice?

***Evaluation:**

- What are the characteristics of linear angles (adjacent and supplementary) ?
- What do you observe about the angle measures of the linear angles?

***Question Corner:**

- What do adjacent , alternate, linear , corresponding and consecutive mean individually
- What are complementary angles?
- What are supplementary angles ?
- What does it mean if two angles are congruent?
- What is the complement of 65 degrees
- What is the supplement of 70 degrees?
- Compare angle relationships formed by parallel lines vs. angle relationships formed by non-parallel lines.

**Activity No 2 Angles formed when a transversal intersects parallel lines**

**Estimated Time**: 90 minutes**Materials/ Resources needed**

Laptop, geogebra file, projector and pointer.

**Prerequisites/Instructions, if any**

- The students should have prior knowledge of parallel lines , transversal, angles and types of angles formed when a pair of parallel lines are intersected by a transversal.
- They should know what the terms interior, exterior, adjacent, alternate, consecutive, congruent, linear and corresponding mean.
- Students should know the definition of complementary angles, supplementary angles, and congruent angles.

**Multimedia resources**:

Laptop

**Website interactives/ links/ / Geogebra Applets**

**Process**

- Reiterate that when a transversal intersects parallel lines, several pairs of congruent and supplementary angles are formed.
- Have students draw two parallel lines and a third line(transversal) intersecting those two lines on their own paper. Direct them to think about any angle relationships they see. Have them discuss their conjectures with a partner.
- The teacher can next project the GeoGebra worksheet and discuss about types of angles and their relationships with the class .
- Finally the teacher and students can summarize together the angle relationships.

Linear pair of angles - adjacent and supplementary

Vertical angles - congruent

Corresponding angles -congruent

Alternate interior angles - congruent

Same side interior angles - supplementary

Alternate exterior angles - congruent

Same side exterior angles - supplementary

**Developmental Questions**

- How many pairs of corresponding angles are there ?
- What is true about corresponding angles formed when parallel lines are cut by a transversal?
- Compare different pairs of alternate interior angles. What do you notice?
- <EGD and <AHF are alternate exterior angles. What is another pair of alternate exterior angles?
- Compare different pairs of same-side interior angles. What do you notice?
- Compare different pairs of same-side exterior angles. What do you notice?

**Evaluation**

- What are the characteristics of linear angles (adjacent and supplementary) ?
- What do you observe about the angle measures of the linear angles?

**Question Corner**

- What do adjacent , alternate, linear , corresponding and consecutive mean individually
- What are complementary angles?
- What are supplementary angles ?
- What does it mean if two angles are congruent?
- What is the complement of 65 degrees
- What is the supplement of 70 degrees?
- Compare angle relationships formed by parallel lines vs. angle relationships formed by non-parallel lines.

# Hints for difficult problems

# Project Ideas

# Math Fun

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