Question

angle pairs formed by a transversal
math
a) name the relation between the angles formed by the transversal in the
figure below.
∠1 and ∠8 =
∠4 and ∠6 =
∠4 and ∠5 =
∠2 and ∠6 =
∠3 and ∠5 =
∠3 and ∠6 =
∠1 and ∠5 =
∠2 and ∠7 =

b) use your knowledge of angle pairs to find the measures of the specified
angles in the following.
①
∠x =____ ∠y =____
②
∠x =____ ∠y =____
③
∠x =____ ∠y =____
④
∠x =____ ∠y =____

Explanation:

Step1: Identify vertical angles

Vertical angles are equal. In the first sub - question (1), \(\angle x\) and the \(30^{\circ}\) angle are vertical angles. So \(\angle x = 30^{\circ}\).

Step2: Use the property of same - side interior angles

For \(\angle y\) in (1), \(\angle y\) and \(\angle x\) are same - side interior angles. Since the sum of same - side interior angles is \(180^{\circ}\), \(\angle y=180^{\circ}-\angle x\). Substituting \(\angle x = 30^{\circ}\), we get \(\angle y = 150^{\circ}\).

Step3: For (2), use the property of alternate interior angles

\(\angle x\) and the \(127^{\circ}\) angle are alternate interior angles. So \(\angle x = 127^{\circ}\). \(\angle y\) and \(\angle x\) are vertical angles. So \(\angle y=\angle x = 127^{\circ}\).

Step4: For (3), use the property of vertical angles and linear pairs

\(\angle x\) and the \(64^{\circ}\) angle are vertical angles. So \(\angle x = 64^{\circ}\). \(\angle y\) and the \(64^{\circ}\) angle form a linear pair. Since the sum of angles in a linear pair is \(180^{\circ}\), \(\angle y=180^{\circ}-64^{\circ}=116^{\circ}\).

Step5: For (4), use the property of vertical angles and linear pairs

\(\angle x\) and the \(73^{\circ}\) angle are vertical angles. So \(\angle x = 73^{\circ}\). \(\angle y\) and \(\angle x\) form a linear pair. Since the sum of angles in a linear pair is \(180^{\circ}\), \(\angle y=180^{\circ}-73^{\circ}=107^{\circ}\).

Answer:

1. \(\angle x = 30^{\circ}\), \(\angle y = 150^{\circ}\)
2. \(\angle x = 127^{\circ}\), \(\angle y = 127^{\circ}\)
3. \(\angle x = 64^{\circ}\), \(\angle y = 116^{\circ}\)
4. \(\angle x = 73^{\circ}\), \(\angle y = 107^{\circ}\)