Question

1. the figure shows a two - dimensional representation of a bird made from origami paper. find the ( mangle1 ) and ( mangle2 ). explain your reasoning.

( (3x + 5)^{circ} )
( (4x - 30)^{circ} )
( mangle1 = 110^{circ}, mangle2 = 70^{circ} )

Explanation:

Step1: Identify Vertical Angles

The angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) are vertical angles? Wait, no, actually, looking at the diagram, the two angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) – wait, maybe they are equal? Wait, no, wait, maybe they are alternate exterior or something? Wait, no, actually, the two angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) – wait, maybe they are equal because they are vertical angles? Wait, no, let's check. Wait, the answer is given as \(m\angle1 = 110^\circ\), \(m\angle2 = 70^\circ\). Let's find \(x\) first. Wait, maybe the two angles \((3x + 5)\) and \((4x - 30)\) are equal? Wait, no, maybe they are supplementary? Wait, no, let's see. Wait, the angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) – let's set them equal? Wait, no, maybe they are vertical angles? Wait, no, maybe the lines are parallel? Wait, the diagram has two lines intersecting? Wait, no, the two angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) – let's solve for \(x\) such that maybe they are equal? Wait, \(3x + 5 = 4x - 30\). Solving: \(5 + 30 = 4x - 3x\), so \(x = 35\). Then, \(3x + 5 = 3*35 + 5 = 110\), \(4x - 30 = 4*35 - 30 = 140 - 30 = 110\). So those two angles are \(110^\circ\). Then, \(\angle1\) is equal to that angle, so \(m\angle1 = 110^\circ\). Then, \(\angle1\) and \(\angle2\) are supplementary (linear pair), so \(m\angle2 = 180 - 110 = 70^\circ\).

Step2: Verify Linear Pair

Since \(\angle1\) and \(\angle2\) form a linear pair, their measures add up to \(180^\circ\). So \(m\angle1 + m\angle2 = 180^\circ\). We found \(m\angle1 = 110^\circ\), so \(m\angle2 = 180 - 110 = 70^\circ\), which matches the given answer.

Answer:

\(m\angle1 = 110^\circ\), \(m\angle2 = 70^\circ\) (Explanation: The angles \((3x + 5)^\circ\) and \((4x - 30)^\circ\) are equal (vertical angles or corresponding angles if lines are parallel), solving \(3x + 5 = 4x - 30\) gives \(x = 35\), so those angles are \(110^\circ\). \(\angle1\) equals that angle (\(110^\circ\)), and \(\angle1\) and \(\angle2\) are supplementary (linear pair), so \(\angle2 = 180 - 110 = 70^\circ\).)