Question

look at the diagram. which equation can be used to solve for x? 6x - 6 = 180, 12x = 90, 12x = 180, 6x - 6 = 90. solve for x.

Explanation:

Part 1: Find the equation to solve for \( x \)

Step1: Identify vertical angles

Angles \( \angle ECG = 2x^\circ \) and \( \angle FCD=(4x - 6)^\circ \) are not vertical angles. Wait, actually, the angle between \( CG \) and \( CD \) is \( 90^\circ \)? No, wait, \( BC \) and \( CD \) are a straight line? Wait, no, \( BC \perp CG \) (right angle at \( C \) between \( BC \) and \( CG \)). Also, \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \), and since \( \angle ECG \) and \( \angle FCD \) are related to the right angle? Wait, no, actually, the sum of \( \angle ECG \), the right angle, and \( \angle FCD \)? Wait, no, looking at the diagram, \( BC \) and \( CD \) are a straight line (vertical line), \( CG \) is horizontal, so \( \angle BCG = 90^\circ \). Also, \( EF \) is a straight line passing through \( C \), so \( \angle ECG \) and \( \angle FCD \) are related? Wait, no, the angle \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \), and also, the angle between \( CG \) and \( CD \) is \( 90^\circ \)? Wait, no, \( BC \) is vertical, \( CG \) is horizontal, so \( \angle BCG = 90^\circ \). Then, the angles \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \), and since \( EF \) is a straight line, the sum of \( \angle ECG \), \( \angle BCG \) (90 degrees), and \( \angle FCD \)? No, wait, actually, \( \angle ECG \) and \( \angle FCD \) are vertical angles? No, vertical angles are equal. Wait, no, \( \angle ECB \) and \( \angle FCD \)? Wait, maybe I made a mistake. Let's re - examine:

The right angle at \( C \) (between \( BC \) and \( CG \)) means \( \angle BCG = 90^\circ \). Also, the line \( EF \) intersects the vertical line \( BD \) at \( C \). The angle \( \angle ECG = 2x \) and the angle \( \angle FCD=(4x - 6) \). Also, since \( \angle ECG \) and \( \angle FCD \) are related to the right angle? Wait, no, the sum of \( \angle ECG \), \( \angle BCG \) (90°), and \( \angle FCD \) is not correct. Wait, actually, the angle between \( CG \) and \( CD \) is \( 90^\circ \), and the angles \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \) are such that \( 2x+(4x - 6)+90 = 180 \)? No, that's not right. Wait, no, the straight line \( BD \) (vertical) and the straight line \( EF \) intersect at \( C \). The angle \( \angle BCG = 90^\circ \), so the sum of \( \angle ECG \) (2x) and \( \angle FCD \) (4x - 6) and the right angle? Wait, no, the correct approach: since \( BC \perp CG \), \( \angle BCG = 90^\circ \). Also, the angles \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \), and since \( EF \) is a straight line, the sum of \( \angle ECG \), \( \angle BCG \), and \( \angle FCD \) is not. Wait, actually, the angle \( \angle ECD \) is a straight line? No, \( BD \) is a straight line, \( EF \) is a straight line. The intersection of \( BD \) and \( EF \) at \( C \) creates vertical angles. Also, \( \angle BCG = 90^\circ \), so \( \angle ECG + \angle FCD=90^\circ \)? Wait, no, \( 2x+(4x - 6)=90 \)? Let's check: \( 2x + 4x-6=90 \), which simplifies to \( 6x - 6 = 90 \). Let's verify the other options:

• Option 1: \( 6x - 6 = 180 \): If we assume the sum is 180, but we have a right angle, so the sum of these two angles should be 90, not 180.
• Option 2: \( 12x = 90 \): Doesn't match the angle relationship.
• Option 3: \( 12x = 180 \): Also doesn't match.
• Option 4: \( 6x - 6 = 90 \): This comes from \( 2x+(4x - 6)=90 \) (since the two angles \( 2x \) and \( 4x - 6 \) are complementary to the right angle? Wait, no, actually, since \( BC \perp CG \), the angle between \( CG \) and \( CD \) is \( 90^\circ \), and the angles \( \angle ECG = 2x \) and \( \angle FCD=(4x - 6) \) are such…

Step1: Start with the equation

We have the equation \( 6x - 6 = 90 \).

Step2: Add 6 to both sides

Add 6 to both sides of the equation: \( 6x-6 + 6=90 + 6 \), which simplifies to \( 6x=96 \).

Step3: Divide by 6

Divide both sides by 6: \( x=\frac{96}{6}=16 \).

Answer:

(Equation): \( 6x - 6 = 90 \)