Question

How would I prove that $\Delta APK$ is congruent to $\Delta RPK$ and it is given that sides AP and RK are equal and $\angle APK\And \angle RKP$ are also equal?

Answer 1

Hint: In the given figure, it is given that we have one pair of sides as equal and also the two angles in the two triangles are equal. And in $\Delta APK$ and $\Delta RPK$, side PK is common in both the two triangles so using SAS congruence we can show that $\Delta APK$ is congruent to $\Delta RPK$.

Complete step by step solution:
The figure given in the above problem is as follows:

In this figure and in $\Delta APK$ and $\Delta RPK$, one pair of sides are equal which is shown below in the following way: $AP=RK$ ……. (1)
And the two angles corresponding to the two triangles are also equal and given as:
$\angle APK=\angle RKP$ ………… (2)
Also, you can see that side PK is common in both the triangles. This means that one of the three sides in both the triangles are equal.
Now, as you can see that one side, one angle and also one side of one triangle is equal to the other side, angle and other side of the triangle. Hence, by SAS congruence we can say that:
$\Delta APK\cong \Delta RPK$
Hence, we have proved that $\Delta APK$ is congruent to $\Delta RPK$.

Note: In the above problem, we have proved the congruence of two triangles $\Delta APK$ and $\Delta RPK$ using SAS congruence. There are other congruence rules also which include ASA, SSS congruence. The congruence rule is easy to understand because SSS denotes that all the three sides of two triangles are equal and ASA denotes one angle, one side and one angle of the two triangles are equal. So, in some problems you might need to use these congruence rules (which are SSS, ASA).