Similar and congruent triangles

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Teaching Outlines

Concept # 1. Congruent triangles

Learning objectives

1. Analyse and identify the structure of simple triangles
2. Gather information about the similarities and differences between triangles
3. Comprehend the meaning of congruent triangles - Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
4. Utilise the newly acquired knowledge in order to solve related problems.
5. Ability to draw congruent triangles
6. List four basic properties of congruent triangles.
7. Properties of Congruent Triangles:
   

• If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.

i.e. CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent".

• In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.

Notes for teachers

1. The teacher can ask students to think of many objects which are mass-produced and that are found to be exactly the same size and shape like pens, CD-roms, cars etc.

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Concept # 2. Tests for congruency of triangles.

Learning objectives

Any triangle is defined by six measures (three sides, three angles). Triangles are congruent if:

1. All three corresponding sides are equal in length. SSS (side side side) congruency postulate
2. A pair of corresponding sides and the included angle are equal. -- SAS (side angle side) congruency postulate.
3. A pair of corresponding angles and the included side are equal. -- ASA (angle side angle) congruency postulate.
4. A pair of corresponding angles and a non-included side are equal.-- AAS (angle angle side) congruency postulate.
5. HL (hypotenuse leg of a right triangle) :Two right triangles are congruent if the hypotenuse and one leg are equal.Also known as RHS postulate.

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Concept # 3. Theorms on congruent triangles

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Concept # 4.Similar triangles

Learning objectives

1. To develop an intuitive understanding of the concept “similarity of figures”.
2. Triangles are similar if they have the same shape, but can be different sizes.
3. Understand that 'corresponding' means matching and 'congruent' means equal in measure.
4. To determine the correspondences between the pairs of similar triangles.
5. The ratio of the corresponding sides is called the ratio of simultude or scale factor.
6. Triangles are similar if their corresponding angles are congruent and the ratio of their corresponding sides are in proportion.
7. To develop an ability to state and apply the definition of similar triangles.
8. recognize and apply “corresponding sides of similar triangles are proportional”.

Notes for teachers

1. The teacher can bring different sized photographs got from same negative like stamp size, passport size and a post card size .
2. Compare them and say that all photos are look alikes and are proportionate. only the size differs.
3. She can also mention about scale concept in graphical representation.
4. Hence similar triangles are the same proportionate triangles but of different sizes.
5. Two triangles are similar if they have:

• all their angles equal
• corresponding sides in the same ratio

1. In similar triangles, the sides facing the equal angles are always in the same ratio. Application of this finds its use in finding the unknown lengths in similar triangles . For this :
   

Step 1: Find the ratio of corresponding sides in pairs of similar triangles.
Step 2: Use that ratio to find the unknown lengths.

Activity No # 1. SIMILAR TRIANGLES

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• Estimated Time:45 minutes.
• Materials/ Resources needed:

Laptop, geogebra file, projector and a pointer.

• Prerequisites/Instructions, if any:

1. The students should have prior knowledge of triangles , sides , angles , vertices .
2. They should know meaning of the terms 'similar' and 'proportionate'.
3. They should be able to identify the corresponding sides.
4. They should know how to find ratio.
5. They should know to find area and perimeter of triangles.

• Multimedia resources: Laptop
• Website interactives/ links/ / Geogebra Applets
• Process:

• Developmental Questions:

1. Look at the shape of both triangles being formed? (look alikes )
2. As I increase /decrease the size of triangles do you see that the measures are changing proportionately ?
3. Can any one explain what exactly proportionately means ?
4. Can you identify the corresponding sides and angles ?

• Evaluation:

1. Name the corresponding sides.
2. Compare the perimeters of two similar triangles.
3. What are equiangular triangles ?

• Question Corner:

1. Compare the ratio of corresponding sides of similar triangles. What do you infer ?
2. How can one draw similar triangles if only one triangles sides are given ?
3. Discuss the applications of similar triangles in finding unknowns in real life situations.
4. Give examples where one uses the concept of similarity.

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Concept # 5. Similarity postulates

Learning objectives

Two triangles are said to be similar if any of the following equivalent conditions hold:

1. The SSS similarity postulate states that if the sides of two triangles are in proportion, then the triangles are similar.
2. The AA similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are said to be similar.
3. SAS Similarity Postulate states, “If an angle of one triangle is congruent to the corresponding angle of another triangle and the sides that include this angle are proportional, then the two triangles are similar.”

Activity No # Similarity test (AA postulate)

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• Estimated Time :45 minutes
• Materials/ Resources needed: Laptop, geogebra file, projector and a pointer
• Prerequisites/Instructions, if any

1. The students should know the meaning of the terms congruent and similar.
2. They should understand the terms corresponding sides and angles.
3. They should have an idea of ratio and proportion.

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

• Process:

1. The teacher can initially have a warm up session regarding terms congruence, similarity and corresponding angles and ratio.
2. She can then project the geogebra file and by moving the sliders she can change the side and angle measures and teach teh AA similarity postulate.
3. Also she can let them understand that in similar triangles, the corresponding sides are proportional.

• Developmental Questions:

1. What does congruent mean ?
2. What does similarity mean ?
3. How can we test whether the two given figures are similar or not ?
4. In the above two triangles, what measures of both are same ?
5. Identify the corresponding sides and angles.
6. Is their ratio same ?
7. What can you say about the two triangles ?
8. Recall the similarity postulates.
9. By what postulate are the two triangles similar ?

• Evaluation:

1. Differentiate similarity and congruence.

• Question Corner:

1. Can the tree and its shadow be considered as similar figures ?
2. Can this similarity concept be used to find the height and depth of objects ? Frame any two of your own questions which can be solved using similarity postulates.

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Concept # 6. Theorms on Similar triangles

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