Question

calculate the size of angle \\(npq\\).

Explanation:

⚡ Using what you learned: lines and angles (parallel, transversal, vertical)

Step 1: Find the angles in triangle RNQ

Triangle \( \text{RNQ} \) is an isosceles triangle because the tick marks show that side \( \text{RN} \) is equal in length to side \( \text{NQ} \) (\( \text{RN} = \text{NQ} \)).

In an isosceles triangle, the angles opposite the equal sides are also equal:

$$ \angle \text{RQN} = \angle \text{NRQ} = 35^\circ $$

The sum of angles in a triangle is \( 180^\circ \):

$$ \angle \text{RNQ} = 180^\circ - (35^\circ + 35^\circ) = 110^\circ $$

Step 2: Use parallel lines to find angle PNQ

The arrows on lines \( \text{NP} \) and \( \text{RQ} \) indicate that they are parallel (\( \text{NP} \parallel \text{RQ} \)).

Using the alternate interior angles theorem with the transversal line \( \text{NQ} \):

$$ \angle \text{PNQ} = \angle \text{RQN} = 35^\circ $$

Step 3: Find angle NQP

We are given that the total angle \( \angle \text{RQP} = 106^\circ \).

Since \( \angle \text{RQP} = \angle \text{RQN} + \angle \text{NQP} \):

$$ \angle \text{NQP} = 106^\circ - 35^\circ = 71^\circ $$

Step 4: Calculate angle NPQ

In triangle \( \text{NPQ} \), the sum of the interior angles must be \( 180^\circ \):

$$ \angle \text{NPQ} + \angle \text{PNQ} + \angle \text{NQP} = 180^\circ $$

$$ \angle \text{NPQ} + 35^\circ + 71^\circ = 180^\circ $$

$$ \angle \text{NPQ} + 106^\circ = 180^\circ $$

$$ \angle \text{NPQ} = 180^\circ - 106^\circ = 74^\circ $$

Answer:

\( 74^\circ \)