Rudransh has a piece of cardboard in shape of a quadrilateral.
He cuts it into two triangles such that angles formed at the
vertices are bisected. He wants to find out whether the two
triangles are congruent or not without actually super positioning
them on one another.
I'm pretty sure that such a quadrilateral would have to be a rhombus. In that case, the triangles would be congruent (SSS).
To determine whether the two triangles formed by bisecting a quadrilateral are congruent or not without superimposing them, we can examine their corresponding sides and angles.
1. Start by drawing the given quadrilateral on a piece of paper or a digital drawing tool. Label each vertex with A, B, C, and D, respectively.
2. Next, draw the line segments connecting the midpoints of opposite sides of the quadrilateral. Label the midpoint of AB as E, BC as F, CD as G, and DA as H.
3. As the quadrilateral was divided into two triangles by bisecting the angles at E, F, G, and H, we need to determine the corresponding sides and angles of these two triangles to check for congruence.
Triangle AEF corresponds to triangle CHF.
Triangle BFG corresponds to triangle DHG.
4. To check for congruence, we can compare the corresponding sides and angles:
a) Corresponding sides:
- Compare side AE with side CH.
- Compare side AF with side CF.
- Compare side BE with side DH.
- Compare side BG with side DG.
If all the corresponding sides are equal, it indicates that the two triangles are congruent.
b) Corresponding angles:
- Compare angle AEF with angle CHF.
- Compare angle BFG with angle DHG.
If all the corresponding angles are equal, it also indicates that the two triangles are congruent.
5. After comparing the corresponding sides and angles, if the lengths of the corresponding sides and measures of the corresponding angles are equal in both triangles, then we can conclude that the two triangles are congruent.
Remember, congruent triangles have equal corresponding sides and angles.