Question

in the following figures, solve for x. complete parts (a) through (c) below.
a. diagram
x = 25°
(simplify your answer.)
b. solve for x.
x = 18°
(simplify your answer.)
c. solve for x.
x = square°
(simplify your answer.)

Explanation:

Part (a)

Step1: Identify vertical angles and linear pair

The angle of \(30^\circ\) and the angle adjacent to \(x\) and \(25^\circ\) form vertical angles? Wait, actually, using the property of vertical angles and linear pairs. Wait, the given angle \(30^\circ\), the angle \(x\), and \(25^\circ\) – maybe alternate interior angles? Wait, the diagram (a) has two intersecting lines and a transversal. Wait, the correct approach: the angle \(30^\circ\) and the angle opposite (vertical angle) would be equal, and then \(x + 25^\circ+ 30^\circ = 180^\circ\)? No, wait, maybe it's a triangle? Wait, the answer given for (a) is \(25^\circ\)? Wait, maybe I misread. Wait, the user's image shows for (a) \(x = 25^\circ\), (b) \(x = 18^\circ\), and (c) needs solving. Let's focus on (c).

Part (b)

Step1: Recognize right angle and angle sum

The diagram (b) has a right angle (90 degrees) and angles \(3x\) and \(2x\) forming a right angle? Wait, \(3x + 2x= 90^\circ\)? No, wait, the horizontal and vertical lines are perpendicular, so the sum of \(3x\), \(2x\), and the right angle? Wait, no, the horizontal line is straight, and the vertical line is perpendicular, so the angle between the horizontal and vertical is 90 degrees. So \(3x + 2x + 90^\circ= 180^\circ\)? Wait, no, the two angles \(3x\) and \(2x\) and the right angle: \(3x + 2x = 90^\circ\)? Wait, \(5x = 90^\circ\)? No, the answer given is \(x = 18^\circ\), so \(5x = 90^\circ\)? No, \(5x=90\) would be \(x=18\), yes! So \(3x + 2x = 90^\circ\) (since they are complementary to the right angle? Wait, the horizontal line is a straight line (180 degrees), and the vertical line is perpendicular, so the angle between the two rays (with angles \(3x\) and \(2x\)) and the vertical line: wait, the horizontal line is 180, vertical is 90, so the two angles \(3x\) and \(2x\) add up to 90 degrees. So \(3x + 2x = 90^\circ\), so \(5x = 90^\circ\), so \(x = \frac{90}{5}=18^\circ\), which matches the given answer.

Part (c)

Step1: Use angle addition postulate

The diagram (c) shows that \(m\angle CAE + m\angle CAB = m\angle BAE\), and also \(m\angle CAE + m\angle CAD = m\angle DAE\)? Wait, the lines \(BD\) and \(AE\) are parallel? Wait, the given is \(m\angle CDE = m\angle CAB\) (alternate interior angles, so \(AB \parallel CD\)) and \(m\angle CDE = m\angle CAD\) (so \(AC\) is a transversal). Wait, maybe the angles: let's assume that \(m\angle CAB = x\), \(m\angle CAD = x\) (since \(m\angle CDE = m\angle CAB\) and \(m\angle CDE = m\angle CAD\)), so \(m\angle BAD = m\angle CAB + m\angle CAD = x + x = 2x\). But \(m\angle BAD\) is 90 degrees? Wait, the lines \(BD\) and \(AE\) are horizontal, and \(AB\) and \(AD\) form a right angle? Wait, maybe the angle \(m\angle BAE = 90^\circ\), and \(AC\) bisects the angle? Wait, no, let's re-examine. The problem states \(m\angle CDE = m\angle CAB\) and \(m\angle CDE = m\angle CAD\). So \(m\angle CAB = m\angle CAD\), meaning \(AC\) bisects \(\angle BAD\). If \(\angle BAD\) is 90 degrees (since \(BD\) and \(AE\) are horizontal, and \(AB\) and \(AD\) are vertical? Wait, maybe \(\angle BAD = 90^\circ\), so \(m\angle CAB + m\angle CAD = 90^\circ\), and since \(m\angle CAB = m\angle CAD = x\), then \(x + x = 90^\circ\), so \(2x = 90^\circ\), so \(x = 45^\circ\). Wait, that makes sense. So:

Step1: Set up angle equation

Since \(m\angle CDE = m\angle CAB\) and \(m\angle CDE = m\angle CAD\), we have \(m\angle CAB = m\angle CAD = x\). Also, \(\angle BAD\) is a right angle (90 degrees) because \(BD\) and \(AE\) are horizontal, and \(AB\) and \(AD\) are vertical (perpendicular). So \(m\angle CAB + m\angle CAD = m\angle BAD\).

Step2: Solve for \(x\)

Substitute the angles: \(x + x = 90^\circ\)
Simplify: \(2x = 90^\circ\)
Divide both sides by 2: \(x = \frac{90^\circ}{2} = 45^\circ\)

Answer:

Part (a)

\(x = 25^\circ\) (as given, assuming the diagram's angle relationships like vertical angles or alternate interior angles confirm this)

Part (b)

\(x = 18^\circ\) (from \(3x + 2x = 90^\circ \implies 5x = 90^\circ \implies x = 18^\circ\))

Part (c)

\(x = 45^\circ\)