Names: Ruei Leng (8) and Evelyn (12) Class: 2E3          Date:14/7/17 

Objective: To investigate counterexamples that show two triangles cannot be congruent.

George said that all equilateral triangles are congruent.

John said that two triangles are congruent if only two pairs of corresponding sides and a pair of non-included angles are equal.

a)Do you agree with their statements?

b)Explain with appropriate diagrams and working

a)We do not agree with both statements.

b)

We do not agree with George because not all equilateral triangles are congruent because two triangles which are equilateral could be of different size and thus are not congruent. 

Even though both triangles in the picture above are equilateral, they are not of the same size and thus they are similar and not congruent triangles. The triangles need to have the same lengths of the sides, same angles and same size to be congruent.

We disagree with John because even if two triangles have two pairs of corresponding sides and a pair of non-included angle they are not necessarily congruent. From the picture below, even though the sides AB=A'B', AC=A'C' and angle ABC=A'B' angle , the two triangles are obviously not congruent. 

Jing Jie (21)

Kah Siang (31)

George said that all equilateral triangles are congruent.

Qn (a) Do you agree or disagree with their statement

a)  We disagree with George’s statement.

Qn (b) Explain with appropriate diagrams and working

b) Two equilateral triangles might not necessarily be equal as congruent figures are the same size and angles for corresponding sides. However, equilateral triangles have the same angle (60॰) only but might not have the same sides. For example, one equilateral triangle might have 1cm sides while another might have 2cm sides. By definition: 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. In the above example, there are two equilateral triangles which are similar but they are not congruent as the sides are not the same although they are both equilateral triangles and have the same angles on all sides. The definition of similar is having the same angles but having different sides but also must be to a fixed ratio. Therefore, equilateral triangles are definitely similar but not definitely congruent.

John said that two triangles are congruent if only two pairs of corresponding sides and a pair of non-included angle are equal.

a) We
 disagree with John’s statement.

b) The triangles cannot necessarily be congruent if the law of side-side-angle applies. Given two sides and an angle like the below given figure, it will not work as we cannot solve find out side a and angle A, The side a and angle A might be different even though given side AC and AB and angle B. If given angle A and the two sides, the triangles can be congruent but if given angle B only with the two other sides, it will not necessarily be congruent. The side AC might be more diagonal and make angle C and A different thus making it not congruent.

George: All equilateral triangles are similar but not congruent. A property of equilateral triangles includes that all of their angles are equal to 60 degrees. If all of the angles in a triangle are equal to the corresponding angles in another triangle, then those two triangles are equal. Since every equilateral triangle's angles are 60 degrees, every equilateral triangle is similar to one another. Whereas for a figure to be congruent, it need to have both the same shape and size as tho other figure thus proving George statement wrong.

John: Two figures can only be congruent only if they have the same shape, size or having the same mirror image to each other. However, John said that two triangles can only be congruent if they have corresponding sides and a pair of non-included angle which is not true as there was no mention about them having the same size, shape or mirror image. Thus this proves that his statement is too vague and cannot confirm that the two triangles are congruent.

Sorry but i had to upload again as i could not delete the previous post