Question

1. \\(a = 2.4\\) and \\(b = 9\\)

1. \\(x = 6\\) and \\(y = 3.5\\)

Explanation:

Evaluate opposite sides of PQRS

Using the Parallelogram Theorems and Solving Linear Equations knowledge points

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Evaluate consecutive angles of PQRS

Using the Parallelogram Theorems and Solving Linear Equations knowledge points

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Since consecutive angles are supplementary, \(QR \parallel PS\). With one pair of opposite sides equal (\(PQ = RS\)) and one pair of sides parallel (\(QR \parallel PS\)), this is an isosceles trapezoid, not necessarily a parallelogram.

Evaluate opposite sides of JKLM

Using the Parallelogram Theorems and Solving Linear Equations knowledge points

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Evaluate other opposite sides of JKLM

Using the Parallelogram Theorems and Solving Linear Equations knowledge points

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Since both pairs of opposite sides are equal (\(JM = KL\) and \(JK = ML\)), the quadrilateral JKLM is a parallelogram.

Answer:

Question 2

The given values do not guarantee that quadrilateral \(PQRS\) is a parallelogram.

Question 3

The given values prove that quadrilateral \(JKLM\) is a parallelogram because both pairs of opposite sides are equal:

$$ JM = KL = 14 \quad \text{and} \quad JK = ML = 22 $$