Topicm6897b84810f024be_1528449000663_0Topic

Characteristics of congruent trianglescongruent trianglescongruent triangles

Levelm6897b84810f024be_1528449084556_0Level

Second

Core curriculumm6897b84810f024be_1528449076687_0Core curriculum

VIII. The properties of planar geometric figures. The student:

4) knows and uses the characteristics of congruent trianglescongruent trianglescongruent triangles.

Timingm6897b84810f024be_1528449068082_0Timing

45 minutes

General objectivem6897b84810f024be_1528449523725_0General objective

Noticing regularities, similarities and analogies and formulating relevant conclusions 

Specific objectivesm6897b84810f024be_1528449552113_0Specific objectives

1. Discovering the characteristics of congruent trianglescongruent trianglescongruent triangles.

2. Using the characteristics of congruent triangles.

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

Learning outcomesm6897b84810f024be_1528450430307_0Learning outcomes

The student:

- discovers the characteristics of congruent triangles,

- uses the characteristics of congruent trianglescongruent trianglescongruent triangles.

Methodsm6897b84810f024be_1528449534267_0Methods

1. Brainstorming.

2. Situational analysis.

Forms of workm6897b84810f024be_1528449514617_0Forms of work

1. Individual work.

2. Group work

Lesson stages

Introductionm6897b84810f024be_1528450127855_0Introduction

By brainstorming, the students revise:

- what figures are called congruent,

- what properties the pairs of congruent figures have,

- what is the relation between the sides and angles in congruent trianglescongruent trianglescongruent triangles.

Task
Revise, what figures are called congruent figures. What kind of properties they have?

Task
Find the image of ABC triangle in axial symmetry with respect to axis p.

Are ABC triangletriangletriangle and its image congruent?

Task
Find the image of ABC triangle in symmetry about a pointpointpoint with respect to point B.

Are ABC triangletriangletriangle and its image congruent?

Procedurem6897b84810f024be_1528446435040_0Procedure

The teacher informs the students that for identifying two congruent triangles, you do not need to use the definition of congruence, but you can use the characteristics of congruent trianglescongruent trianglescongruent triangles, that is the conditions that guarantee their congruence.

Task
The teacher divides the class into groups of 4‑5 students and asks them to think through how many sides or angles you need to compare to be sure that triangles are congruent. After 10 minutes’ group discussion, the students and the teacher give the characteristics of congruent triangles.

First characteristic of congruent triangles: side‑side‑side.
If three pairs of sides of two triangles are equal in length, then the triangles are congruent.m6897b84810f024be_1527752263647_0First characteristic of congruent triangles: side‑side‑side.
If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

[Illustration 1]

If |AB| = |DE|, |AC| = |DF|, |CB| = |FE| then ABC triangle and DEF triangletriangletriangle are congruent.

We can write it shortly as:

Second characteristic of congruent triangles: side‑angle‑side.
If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.m6897b84810f024be_1527752256679_0Second characteristic of congruent triangles: side‑angle‑side.
If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

[Illustration 2]

If |AB| = |DE|, |AC| = |DF|, |∡BAC| = |∡EDF| then ABC triangletriangletriangle and DEF triangle are congruent.

Third characteristic of congruent triangles: angle‑side‑angle.
If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.m6897b84810f024be_1527712094602_0Third characteristic of congruent triangles: angle‑side‑angle.
If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

[Illustration 3]

If |AB| = |DE|, |∡BAC| = |∡EDF|, |∡ABC| = |∡DEF| to then ABC triangle and DEF triangle are congruent.

Task
Students work individually, using computers.

Their task is to check if there exists a triangletriangletriangle where two sides and the angleangleangle that is not between this sides are known and it is congruent to the given triangle.

[Geogebra applet]

Having completed the exercise, they present the results of their observations by answering the following questions:

- Can you build a triangle congruent to another triangletriangletriangle with known two sides and the angleangleangle that is not between this sides?

- Is there a characteristics of congruent trianglescongruent trianglescongruent triangles SSA?

Task
PointpointPoint E is the mid‑point of the AB line segmentsegmentsegment. Lines DA and BC are parallel. Prove that triangles ADE and BCE are congruent. What characteristiccharacteristiccharacteristic of congruent trianglescongruent trianglescongruent triangles you must use?

[Illustration 4]

Task
Are presented triangles congruent? If yes, what characteristiccharacteristiccharacteristic proves it?

[Illustration 5]

Conclusions:

Two right‑angled triangles are congruent if they have equal:

- catheuses,

- one catheus and hypotenusehypotenusehypotenuse,

- catheuse and the angleangleangle opposite this cathouse,

- hypotenusehypotenusehypotenuse and one of the acute angles.

Task
There is the right‑angled ABC triangletriangletriangle whose hypotenuse equals 8 cm and the acute angleangleangle equals 40° and the right‑angled DEF triangle whose hypotenusehypotenusehypotenuse equals 8 cm and the acute angleangleangle equals 50°. Are these triangles congruent?

An extra task:
Prove that the area of the ABCD triangletriangletriangle is equal to the area of the AFD triangle.

[Illustration 6]

Lesson summarym6897b84810f024be_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to remember.

First characteristic of congruent trianglesFirst characteristic of congruent trianglesFirst characteristic of congruent triangles: side‑side‑side.

- If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

Second characteristic of congruent trianglesSecond characteristic of congruent trianglesSecond characteristic of congruent triangles: side‑angle‑side.

- If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

Third characteristic of congruent trianglesThird characteristic of congruent trianglesThird characteristic of congruent triangles: angle‑side‑angle.

- If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

Conclusions:

Two right‑angled triangles are congruent if they have equal:

- catheuses,

- one catheus and hypotenusehypotenusehypotenuse,

- catheuse and the angleangleangle opposite this cathouse,

- hypotenuse and one of the acute angles.

Selected words and expressions used in the lesson plan

angleangleangle

characteristiccharacteristiccharacteristic

congruent trianglescongruent trianglescongruent triangles

First characteristic of congruent trianglesFirst characteristic of congruent trianglesFirst characteristic of congruent triangles

hypotenusehypotenusehypotenuse

pointpointpoint

Second characteristic of congruent trianglesSecond characteristic of congruent trianglesSecond characteristic of congruent triangles

segmentsegmentsegment

Third characteristic of congruent trianglesThird characteristic of congruent trianglesThird characteristic of congruent triangles

triangletriangletriangle