Rahul was trying to make a paper toy and started with folding an A4 sized paper along the diagonal. He then noticed that there were two triangles formed. Arjun was saying that the two triangles formed are congruent to each other. But then Rahul noticed that they aren’t superimposing on each other, he says that the triangles aren’t congruent.

What are the conditions for a triangle to be congruent?

Orientation of triangles does matter

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Corresponding sides should be equal

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Corresponding angles should be equal

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Orientation of triangles does not matter

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Solution

The correct options are
B
Corresponding sides should be equal

C
Corresponding angles should be equal

D
Orientation of triangles does not matter

AB is the diagonal of the A4 sheet.

AC, BD are the lengths and AD, BC are the breadths of the A4 sheet.

Consider $Δ A D B$ and $Δ A C B$ ,

AD = BC (Breadth of the A4 sheet)

AC = BD (Length of the A4 sheet)

AB is the common side for both the triangle.

Therefore, $Δ A D B$ and $Δ B C A$ are congruent under SSS congruence criterion.

So, if we flip one of the triangles according to the other triangle’s orientation, it superimposes on the other. Hence, orientation is not important but the corresponding sides and angles are. There should be at least one orientation in which both the triangles superimpose on each other.

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Similar questions

Q. Which of the following statements are true and which of them are false?

(i) All squares are congruent.
(ii) If two squares have equal areas, they are congruent.
(iii) If two figures have equal areas, they are congruent.
(iv) If two triangles are equal in area, they are congruent.
(v) If two sides and one angle of a triangle are equal to the corresponding two sides and angle of another triangle, the triangle are congruent.
(vi) If two angles and any side of a triangle are equal to the corresponding angles and the side of another triangle then the triangles are congruent.
(vii) If three angles of a triangle are equal to the corresponding angles of another triangle then the triangles are congruent.
(viii) If the hypotenuse and an acute angle of a right triangle are equal to the hypotenuse and the corresponding acute angle of another right triangle then the triangle are congruent.
(ix) If the hypotenuse of a right triangle is equal to the hypotenuse of another right triangle then the triangles are congruent.
(x) If two triangles are congruent then their corresponding sides and their corresponding angles are congruent.

Q. There exist two triangles such that there five parts ( i.e. three angles and two sides) are congruent to each other but yet the triangles are not congruent ?

In two right triangles, one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.

Q. In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.

Q. AAS (or SAA) test :
The sum of the measures of angles in a triangle is

180^{\circ}

. Therefore, if two corresponding pairs of angles in two triangles are congruent, then the remaining pair of angles is also congruent. Hence, if two angles and a side adjacent to one of them are congruent with corresponding parts of the other triangle then the condition for ASA test is fulfilled. So the triangles are congruent.

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