Question

to compare \\( \angle pqr \\) and \\( \angle def \\) shown in fig. 5.7, trace \\( \angle pqr \\) on a tracing paper and place it on \\( \angle def \\) such that q falls on e and qr falls on de. since pq lies between de and ef, \\( \angle pqr < \angle def \\).

Explanation:

Step1: Understand the Angle - Comparison Method

When comparing two angles by the super - imposition method, we align one side of the first angle with the corresponding side of the second angle. Here, we align \(QR\) with \(DE\) (by making \(Q\) fall on \(E\)).

Step2: Analyze the Position of the Non - Aligned Sides

After aligning \(QR\) (of \(\angle PQR\)) with \(DE\) (of \(\angle DEF\)), we check the position of \(PQ\) (the non - aligned side of \(\angle PQR\)) with respect to \(\angle DEF\). If the non - aligned side of the first angle (\(PQ\)) lies between the two sides (\(DE\) and \(EF\)) of the second angle (\(\angle DEF\)), then the measure of the first angle is less than the measure of the second angle. Mathematically, if we consider the measure of an angle as the amount of rotation from one side to the other, when \(PQ\) is between \(DE\) and \(EF\), the rotation from \(QR\) (now \(DE\)) to \(PQ\) is less than the rotation from \(DE\) to \(EF\).

Answer:

\(\angle PQR<\angle DEF\)