Question

Find the given figure below, show that \[\Delta POS \cong \Delta QOR\].

Answer 1

Hint: Triangle is described as the three sided shape with three intersecting point and the sum of the angles of a triangle is \[180^\circ \]. So, according to the Angle, Angle and Angle congruence and using the alternate angles, opposite angles rule, if the three angles of the two triangles are similar to each other (it means if the angle of one of the angles is \[40^\circ \]then the same angle of the second triangle is same) then we can say that the two triangles are similar in construction so it is easy to show the \[\Delta POS \cong \Delta QOR\]

Complete step-by-step answer:
The angle \[\angle POS = 40^\circ \].
The angle \[\angle PSO = 60^\circ \].
The angle \[\angle QRO = 60^\circ \].
Then, we know that from the two triangles, \[\angle POS = \angle QOR = 40^\circ \] is equal because the two angles are opposite to each and lie on the same lines.
It is known that the sum of the angles of a triangle is \[180^\circ \], then according to the diagram the sum of two angles is given by,
\[60^\circ + 40^\circ = 100^\circ \]
Now, we can say that the third angle will be,
 \[180^\circ - 100^\circ = 80^\circ \]
Since, we know the angle \[\angle SPO = 80^\circ \] and also \[\angle OQR = 80^\circ \] because both are alternate angles.
So, according to the AAA (Angle angle angle) congruence criteria, that is,
\[\angle POS = \angle QOR = 40^\circ \]
\[\angle SPO = \angle OQR = 80^\circ \]
\[\angle PSO = \angle QRO = 60^\circ \]
Now, we can say that the triangle \[\Delta POS\] is similar to \[\Delta QOR\] that is \[\Delta POS \cong \Delta QOR\]..
Therefore, according to the AAA congruence, it is shown that \[\Delta POS \cong \Delta QOR\].

Note: In this solution we used AAA (angle angle angle) process, we showed the two triangles are similar, but we can also use AAS (Angle Angle Side) method to solve this answer, which means, in the two triangles, the two angles will be similar or equal and the third side will be equal in length in the two triangles, so by using this method, we can show that the two triangles will be similar.