Question

use the given figure for 3-5.

1. find the value of z if :

m∠1 = z² + 45°
m∠4 = 55°
z = ______

1. find m∠2.

m∠2 = ______

1. find m∠3.

m∠3 = ______

Explanation:

Question 3

Step1: Identify angle relationship

∠1 and ∠4 are corresponding angles (since lines are parallel, corresponding angles are equal). So \( m\angle1 = m\angle4 \).
Given \( m\angle1 = z^2 + 45^\circ \) and \( m\angle4 = 55^\circ \), we set up the equation:
\( z^2 + 45^\circ = 55^\circ \)

Step2: Solve for \( z^2 \)

Subtract \( 45^\circ \) from both sides:
\( z^2 = 55^\circ - 45^\circ = 10^\circ \)? Wait, no, angles here are in degrees, but \( z^2 \) should be a number (since it's a coefficient for angle measure? Wait, maybe the problem has a typo, but assuming \( m\angle1 = z^2 + 45 \) (without degree symbol for \( z^2 \), as \( z \) is a number, and angle is in degrees). So:
\( z^2 + 45 = 55 \)
\( z^2 = 55 - 45 = 10 \)? No, that can't be. Wait, maybe ∠1 and ∠4 are alternate interior angles? Wait, no, looking at the figure, the two vertical lines are parallel, and the transversal cuts them. So ∠1 and ∠4: wait, maybe ∠1 and ∠4 are equal? Wait, no, maybe ∠1 and ∠2? Wait, no, let's re-examine. Wait, the first vertical line and the transversal: ∠2, ∠3, ∠4. Then the second vertical line and transversal: ∠1. So ∠1 and ∠4: are they corresponding? If the two vertical lines are parallel, then ∠1 and ∠4 are equal (corresponding angles). So \( m\angle1 = m\angle4 \). So \( z^2 + 45 = 55 \)? No, that would give \( z^2 = 10 \), which is not a perfect square. Wait, maybe it's \( m\angle1 = z + 45 \)? Or maybe ∠1 and ∠4 are supplementary? Wait, no, the figure: let's assume that the two vertical lines are parallel, so ∠1 and ∠4: if the transversal is cutting them, then ∠1 and ∠4 are equal (corresponding). Wait, maybe the problem is \( m\angle1 = z^2 + 45 \) and \( m\angle4 = 55 \), but that would be \( z^2 = 10 \), which is not integer. Wait, maybe I made a mistake. Wait, maybe ∠1 and ∠2 are vertical angles? No. Wait, maybe the first vertical line and transversal: ∠2 and ∠4 are vertical? No, ∠2 and ∠4: ∠2 and ∠3 are supplementary, ∠3 and ∠4 are supplementary? No, ∠2 and ∠4 are vertical angles? Wait, no, the intersection of transversal and first vertical line: ∠2, ∠3, ∠4, and the opposite angle of ∠2 is equal to ∠4? Wait, maybe ∠2 and ∠4 are vertical angles, so \( m\angle2 = m\angle4 = 55^\circ \), but that's for question 4. Wait, back to question 3: \( m\angle1 = z^2 + 45 \), \( m\angle4 = 55 \). If ∠1 and ∠4 are equal (corresponding angles, since lines are parallel), then \( z^2 + 45 = 55 \) → \( z^2 = 10 \) → \( z = \sqrt{10} \), but that's not likely. Wait, maybe the problem is \( m\angle1 = z + 45 \), then \( z + 45 = 55 \) → \( z = 10 \). Maybe a typo in the problem, \( z^2 \) should be \( z \). Assuming that, let's proceed.

Step1: Correct the equation (assuming typo, \( z \) instead of \( z^2 \))

\( m\angle1 = z + 45^\circ \), \( m\angle4 = 55^\circ \), and ∠1 = ∠4 (corresponding angles). So:
\( z + 45 = 55 \)

Step2: Solve for \( z \)

Subtract 45 from both sides:
\( z = 55 - 45 = 10 \)

Step1: Identify angle relationship

∠2 and ∠4 are vertical angles? No, ∠2 and ∠4: looking at the figure, ∠2 and ∠4 are adjacent to ∠3. Wait, ∠2 and ∠4: if the transversal intersects the vertical line, then ∠2 and ∠4 are vertical angles? No, ∠2 and ∠4: ∠2 and ∠3 are supplementary, ∠3 and ∠4 are supplementary, so ∠2 = ∠4 (since both supplementary to ∠3). So \( m\angle2 = m\angle4 = 55^\circ \)? Wait, no, ∠4 is 55°, so ∠2 = 55°? Wait, no, ∠2 and ∠4: if ∠2 and ∠4 are vertical angles, then yes. Or ∠2 and ∠1: if lines are parallel, ∠2 and ∠1 are corresponding? Wait, no, ∠1 is at the second vertical line. Wait, ∠2 and ∠4: let's see, the first vertical line and transversal: ∠2, ∠3, ∠4. So ∠2 and ∠4 are vertical angles? No, ∠2 and ∠4: ∠2 is opposite to the angle adjacent to ∠4. Wait, maybe ∠2 and ∠4 are equal because they are vertical angles. So \( m\angle2 = m\angle4 = 55^\circ \)? No, that can't be, because ∠2 and ∠3 are supplementary. Wait, maybe I messed up. Wait, ∠4 is 55°, so ∠3 is 180° - 55° = 125°, and ∠2 is equal to ∠4? No, ∠2 and ∠3 are supplementary, so ∠2 = 180° - ∠3. Wait, no, ∠2 and ∠4: if ∠2 and ∠4 are vertical angles, then they are equal. Wait, the intersection of transversal and vertical line: the four angles are ∠2, ∠3, ∠4, and the angle opposite to ∠2. So ∠2 and ∠4 are vertical angles? No, ∠2 and the angle opposite to ∠4 are vertical. Wait, maybe the figure is such that ∠1 and ∠2 are corresponding angles. Wait, ∠1 is at the second vertical line, ∠2 at the first. So if lines are parallel, ∠1 = ∠2. From question 3, if \( z = 10 \), then \( m\angle1 = 10 + 45 = 55^\circ \), so \( m\angle2 = 55^\circ \). That makes sense. So ∠1 and ∠2 are corresponding angles, so equal.

Step1: Use corresponding angles

Since the two vertical lines are parallel, ∠1 and ∠2 are corresponding angles, so \( m\angle2 = m\angle1 \).

Step2: Substitute \( m\angle1 \)

From question 3, \( m\angle1 = 55^\circ \) (if \( z = 10 \)), so \( m\angle2 = 55^\circ \).

Step1: Identify angle relationship

∠3 and ∠4 are supplementary (they form a linear pair), so \( m\angle3 + m\angle4 = 180^\circ \).

Step2: Substitute \( m\angle4 \)

Given \( m\angle4 = 55^\circ \), so:
\( m\angle3 = 180^\circ - 55^\circ = 125^\circ \)

Answer:

\( z = 10 \) (assuming a typo in the problem, \( z \) instead of \( z^2 \); if \( z^2 \), then \( z = \sqrt{10} \), but likely typo)

Question 4