Question

In the figure given below , which congruence rule can be used to prove $\vartriangle ABC$ $ \cong $ $\vartriangle ADC$ ?

Answer 1

Hint:In the given quadrilateral divide the quadrilateral by the means of diagonal which gives rise to the occurrence of two new triangles $\vartriangle ABC$ and $\vartriangle ADC$ . Now check the side side side congruence for the new triangles $\vartriangle ABC$ and $\vartriangle ADC$

Complete step-by-step answer:
Congruent triangles : Two triangles are said to be congruent if the sides and angles of the two triangles are having equivalent angles and sides
Side Side Side congruency : If all three sides of a triangle are equivalent to the corresponding three sides of another triangle , then the two triangles are said to be in side side side congruency

Here in the above triangle the length of the side PQ is equal to XY and the length of the side PR is equal to the length of the side YZ and length of the side QR is equal to the length of the side XZ
Sides $PQ = XY$, $PR = YZ$, $QR = XZ$.
Since three sides of $\vartriangle PQR$ equivalent to three sides of $\vartriangle XYZ$ , hence these two triangles are in side side side congruency (SSS)

Using this concept we try to solve the question
Now in the question, divide the quadrilateral by the means to diagonal AC which gives rise to occurrence of two new triangles $\vartriangle ABC$ and $\vartriangle ADC$

From the diagram side AC is common in the two triangles and sides and the length of AB is equal to length of DC and the length of BC is equal to length of AD
 $AB = DC$ and $BC = AD$
Since all three sides of the two triangles are equivalent we can conclude it is in side side side congruency

$\therefore $ We proved $\vartriangle ABC$ $ \cong $ $\vartriangle ADC$ by side side side congruency (SSS)

Additional Information:There are totally 5 types of concurrency they are
1) Side-Side-Side concurrency
2) Side-Angle-Side concurrency
3) Angle-Side-Angle concurrency
4) Angle-Angle-Side concurrency
5) Right angle Hypotenuse side
The first four types of concurrency are possible for every triangle but the fifth type of concurrency is only possible for the right angled triangles

Note:If sides are in proportion and angles are equivalent then the triangles are similar not concurrent.Don’t come to a conclusion that sides are equal unless and until it is mentioned in the question by descriptive way or diagrammatic way.