Topicm6ee2f8d0c769c74e_1528449000663_0Topic

Triangles and their properties

Levelm6ee2f8d0c769c74e_1528449084556_0Level

Second

Core curriculumm6ee2f8d0c769c74e_1528449076687_0Core curriculum

IX. Polygons, circles. The student:

1) identifies and names triangles that are: acute, right‑angled, obtuse, equilateral, isosceles;

2) uses the theorems related to the sum of their interior angles.

XI. Calculations in geometry. The student:

1) calculates the area of: the triangletriangletriangle, the square, the rectangle, the rhombus, the parallelogram, the trapezoid, shown in the picture and in practical situations, including data that require the conversion of units and in situations when the dimensions are not typical, for example the area of the triangletriangletriangle with a side of 1 km and the altitude of 1 mm.

Timingm6ee2f8d0c769c74e_1528449068082_0Timing

45 minutes

General objectivem6ee2f8d0c769c74e_1528449523725_0General objective

III. Using and interpreting representation.

1. Using simple, well known mathematical objects, interpreting mathematical concepts and operating on mathematical objects.

Specific objectivesm6ee2f8d0c769c74e_1528449552113_0Specific objectives

1. Naming and identifying various kinds of triangles.

2. Determining the angles in the triangletriangletriangle, constructing the altitude of the triangle.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm6ee2f8d0c769c74e_1528450430307_0Learning outcomes

1. Identifies various kinds of triangles.

2. Determines the angles of the triangle, constructs the altitude of the triangletriangletriangle.

Methodsm6ee2f8d0c769c74e_1528449534267_0Methods

1. Brainstorming.

2. Analysis of a situation.

Forms of workm6ee2f8d0c769c74e_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm6ee2f8d0c769c74e_1528450127855_0Introduction

The teacher introduces the topic of the lesson: revising the classification of triangles concerning their sides and angles as well as the knowledge of the angles and altitudes of triangles.

Task
In the pictures, there are triangles and the descriptions of their interior angles. The students determine the type of each triangle.

[Illustration 1]

Students identify the following triangles: acute, right‑angled, and obtuse.

Procedurem6ee2f8d0c769c74e_1528446435040_0Procedure

Task
Students determine the type of each triangletriangletriangle concerning the lengths of its sides.

Students revise the definition of various triangles: isosceles, equilateral and scalene.

Definition types of triangles:
If two sides of a triangle are the same, is called an isosceles triangle.
A triangle that in which all sides are of the same length is called an equilateral triangle.
A triangle whose all sides are of different length is called a scalene triangle.m6ee2f8d0c769c74e_1527712094602_0If two sides of a triangle are the same, is called an isosceles triangle.
A triangle that in which all sides are of the same length is called an equilateral triangle.
A triangle whose all sides are of different length is called a scalene triangle.

Task
Connect the drawings and the types of the triangles.

Task
Students work individually using their computers. Their task is to observe what the sum of the interior angles of a triangletriangletriangle is.

The students answer the following questions:

Which angle is equal to angleangleangle ∝ ?
What are the names of angles α and δ ?
Which angleangleangle is equal to angle β ?
What are the names of angles β and ε ?
What angleangleangle is made of the sum of angles α, β and γ ?

[Geogebra applet]

The sum of the interior angles.

On the basis of the previously made observations, the students make the generalization.

The sum of the angles of the triangletriangletriangle is 180°.

Task

In an ABC triangle the angle A is 25° and the angle B is 115°. What is the measuremeasuremeasure of the angleangleangle C?

In the next part of the lesson, we will revise the definition and the properties of the altitude of the triangletriangletriangle.

Definition:
The altitude of a triangle is the line segment that connects the apex of the triangle with the line that includes the opposite side of the triangle which is perpendicular to this line. The triangle has three altitudes.m6ee2f8d0c769c74e_1527752256679_0The altitude of a triangle is the line segment that connects the apex of the triangle with the line that includes the opposite side of the triangle which is perpendicular to this line. The triangle has three altitudes.

Definition:
A point at which all the lines that include the altitudes intersect is called the orthocentre of a triangle.m6ee2f8d0c769c74e_1527752263647_0A point at which all the lines that include the altitudes intersect is called the orthocentre of a triangle.

Task
Draw the following triangles: acute, right‑angled, and obtuse. Draw the altitudes in each triangle. Where is the orthocentre of each triangletriangletriangle?

An extra task
In an isosceles triangleisosceles triangleisosceles triangle one of the angles is 30°. Calculate the remaining angles.

Lesson summarym6ee2f8d0c769c74e_1528450119332_0Lesson summary

Students do the revision exercises. Then they together sum‑up the classes by formulating the conclusions to memorise.

We classify the triangles with respect to the angles into: acute, right‑angled, and obtuse. We classify the triangles with respect to the lengths of their sides into isosceles, equilateral and scalene.

The altitude of the triangle is the line segment that connects the apex of the triangletriangletriangle with the line that includes the opposite side of the triangle which is perpendicular to this line. The triangletriangletriangle has three altitudes.

Selected words and expressions used in the lesson plan

altitude of a trianglealtitude of a trianglealtitude of a triangle

angleangleangle

differentdifferentdifferent

measuremeasuremeasure

equilateral triangleequilateral triangleequilateral triangle

isosceles triangleisosceles triangleisosceles triangle

orthocentre of a triangleorthocentre of a triangleorthocentre of a triangle

scalene trianglescalene trianglescalene triangle

triangletriangletriangle

types of trianglestypes of trianglestypes of triangles