Ankita has two triangles \(\Delta ABC\) and \(\Delta PQR\):
The measurements are:
\(AB = PQ = 9.8 \ cm\), \(BC = QR =6.1 \ cm\) and \(\angle A = \angle P = 45^\circ\)
Answer the following questions:
(a) State whether the given angles are included or non-included for the given side
lengths.
lengths.
(b) Can Ankita conclude that \(\Delta ABC \cong \Delta PQR\)?
Explain your reasoning clearly, referring to the geometric construction.
The given measurements correspond to the condition, where the given angle is not included between the two given sides.
Unlike standard congruence conditions (SSS, SAS), the SSA condition does not guarantee a unique triangle.
Geometric Construction:
Step 1 : Draw a base line segment \(AB\) of length \(9.8 \ cm\).
Step 2 : From vertex \(A\), draw a line \(l\) making a fixed angle of \(45^\circ\) with the base \(AB\). This \(45^\circ\) angle is our non-included angle.
Step 3 : Set your compass to a radius of \(6.1 cm\). Place the needle at vertex \(B\) and draw a sufficiently long circular arc cutting through the line \(l\).
Step 4: Observe the intersection. The arc cuts the line \(l\) at two distinct points, \(C\) and \(D\).
Both points \(C\) and \(D\) lie exactly \(6.1 \ cm\) away from vertex \(B\). Thus, we can construct two entirely , triangles using the same three measurements:
1. Triangle 1 \(\Delta ABC\): \(AB = 9.8 \ cm\), \(BC = 6.1 \ cm\), and \(\angle A = 45^\circ\).
2. Triangle 2 \(\Delta ABD\): \(AB = 9.8 \ cm\), \(BD = 6.1 \ cm\), and \(\angle A = 45^\circ\).
This means two non-congruent triangles can be constructed with these identical measurements, so SSA is not sufficient to prove congruence.
Hence, Ankita conclude that \(\Delta ABC \cong \Delta PQR\).