Question

in the figure, g ∥ h and e ∥ f, m∠1 = (2x + 3)°, and m∠3 = (5x + 2)°. what is m∠1?
g h
← 2 1 → e
← 3 → f
m∠1 = °

Explanation:

Step1: Identify angle relationships

Since \( g \parallel h \) and \( e \parallel f \), \( \angle 1 \) and \( \angle 3 \) are equal (corresponding angles in a parallelogram - like figure formed by parallel lines). So, \( 2x + 3 = 5x + 2 \) is incorrect. Wait, actually, \( \angle 2 \) and \( \angle 3 \) are equal (alternate interior angles as \( e \parallel f \) and \( g \) is transversal), and \( \angle 1 \) and \( \angle 2 \) are supplementary? No, wait, \( g \parallel h \) and \( e \parallel f \), so the quadrilateral formed is a parallelogram, so \( \angle 1 \) and \( \angle 3 \) should be equal? Wait, no, let's re - examine. \( e \parallel f \), \( g \) is a transversal, so \( \angle 2 \) and \( \angle 3 \) are alternate interior angles, so \( \angle 2=\angle 3 \). Also, \( g \parallel h \), \( e \) is a transversal, so \( \angle 1 \) and \( \angle 2 \) are same - side interior angles? No, \( g \parallel h \), \( e \) is a transversal, \( \angle 1 \) and \( \angle 2 \) are adjacent angles? Wait, no, \( \angle 1 \) and \( \angle 2 \) are supplementary? Wait, no, the correct relationship: since \( g \parallel h \) and \( e \parallel f \), \( \angle 1 \) and \( \angle 3 \) are equal (because \( \angle 1=\angle 2 \) (vertical angles? No, \( \angle 1 \) and \( \angle 2 \) are same - side? Wait, I made a mistake. Let's start over.

Given \( g\parallel h \) and \( e\parallel f \). \( \angle 1 \) and \( \angle 2 \) are equal (corresponding angles as \( g\parallel h \) and \( e \) is transversal). \( \angle 2 \) and \( \angle 3 \) are equal (alternate interior angles as \( e\parallel f \) and \( g \) is transversal). So, \( \angle 1=\angle 3 \) is wrong. Wait, no, \( \angle 1 \) and \( \angle 2 \) are same - side? No, \( g\parallel h \), \( e \) is a transversal, so \( \angle 1 \) and \( \angle 2 \) are supplementary? No, if \( g\parallel h \) and \( e \) is a transversal, \( \angle 1 \) and \( \angle 2 \) are adjacent angles forming a linear pair? No, the lines \( e \) and \( f \) are parallel, \( g \) and \( h \) are parallel. So \( \angle 1 \) and \( \angle 3 \) are equal because \( \angle 1=\angle 2 \) (vertical angles? No, \( \angle 1 \) and \( \angle 2 \) are actually equal as corresponding angles (since \( g\parallel h \) and \( e \) is transversal). And \( \angle 2=\angle 3 \) (alternate interior angles as \( e\parallel f \) and \( g \) is transversal). So \( \angle 1 = \angle 3 \) is incorrect. Wait, I think I messed up the angle positions. Let's look at the figure: \( e \) and \( f \) are horizontal parallel lines, \( g \) and \( h \) are slanting parallel lines. So \( \angle 1 \) is on line \( e \), between \( h \) and \( e \); \( \angle 3 \) is on line \( f \), between \( g \) and \( f \). So, since \( e\parallel f \) and \( h\parallel g \), \( \angle 1 \) and \( \angle 3 \) are equal (corresponding angles in the parallelogram formed by the four lines). So we set \( 2x + 3=5x - 2 \)? Wait, no, the original problem: \( m\angle 1=(2x + 3)^{\circ} \), \( m\angle 3=(5x + 2)^{\circ} \). Wait, maybe \( \angle 1 \) and \( \angle 3 \) are supplementary? No, that can't be. Wait, no, let's use the correct parallel line angle theorems.

Since \( e\parallel f \) and \( g \) is a transversal, \( \angle 2=\angle 3 \) (alternate interior angles). Since \( g\parallel h \) and \( e \) is a transversal, \( \angle 1+\angle 2 = 180^{\circ} \) (same - side interior angles). So \( \angle 1+\angle 3=180^{\circ} \) (because \( \angle 2 = \angle 3 \)). So \( (2x + 3)+(5x + 2)=180 \).

Step2: Solve for \( x \)

Combine like terms: \( 2x+5x + 3 + 2=180 \)
\( 7x+5 = 180 \)
Subtract 5 from both sides: \( 7x=180 - 5=175 \)
Divide both sides by 7: \( x = 25 \)

Step3: Find \( m\angle 1 \)

Substitute \( x = 25 \) into \( m\angle 1=(2x + 3)^{\circ} \)
\( m\angle 1=2\times25 + 3=50 + 3 = 53^{\circ} \)

Wait, no, that was a mistake. Let's re - do the angle relationship. If \( g\parallel h \) and \( e\parallel f \), then the figure is a parallelogram, so consecutive angles are supplementary? No, in a parallelogram, opposite angles are equal and consecutive angles are supplementary. Wait, maybe \( \angle 1 \) and \( \angle 3 \) are equal. Let's assume \( \angle 1=\angle 3 \) (opposite angles of a parallelogram). Then \( 2x + 3=5x+2 \)
Subtract \( 2x \) from both sides: \( 3 = 3x+2 \)
Subtract 2 from both sides: \( 1 = 3x \)
\( x=\frac{1}{3} \), which gives a non - integer angle, so that's wrong.

Wait, let's look at the figure again. \( e \) and \( f \) are horizontal, \( g \) and \( h \) are slanting downwards to the right. \( \angle 1 \) is on the upper right, \( \angle 3 \) is on the lower left. So \( \angle 1 \) and \( \angle 3 \) are equal because \( e\parallel f \) and \( h\parallel g \), so they are corresponding angles. Wait, maybe the first assumption was wrong. Let's use the correct approach:

Since \( g\parallel h \) and \( e\parallel f \), the quadrilateral formed by the intersection of these lines is a parallelogram. In a parallelogram, \( \angle 1=\angle 3 \) (opposite angles). So:

\( 2x + 3=5x+2 \)

\( 3 - 2=5x - 2x \)

\( 1 = 3x \)

\( x=\frac{1}{3} \), which is not possible. So my angle relationship is wrong.

Wait, \( \angle 2 \) and \( \angle 3 \) are alternate interior angles ( \( e\parallel f \), \( g \) is transversal), so \( \angle 2=\angle 3 \). \( \angle 1 \) and \( \angle 2 \) are supplementary ( \( g\parallel h \), \( e \) is transversal, same - side interior angles). So \( \angle 1+\angle 2 = 180^{\circ} \), and \( \angle 2=\angle 3 \), so \( \angle 1+\angle 3 = 180^{\circ} \)

So \( (2x + 3)+(5x + 2)=180 \)

\( 7x+5 = 180 \)

\( 7x=175 \)

\( x = 25 \)

Then \( m\angle 1=2\times25 + 3=53^{\circ} \)

Yes, that makes sense. So the correct steps are:

Step1: Determine angle sum relationship

Since \( g\parallel h \) and \( e\parallel f \), \( \angle 1 \) and \( \angle 3 \) are same - side interior angles (or supplementary as they are consecutive angles in a parallelogram - like figure), so \( m\angle 1+m\angle 3 = 180^{\circ} \).

Step2: Set up equation

\( (2x + 3)+(5x + 2)=180 \)

Step3: Solve for \( x \)

Combine like terms: \( 7x+5 = 180 \)

Subtract 5 from both sides: \( 7x=175 \)

Divide by 7: \( x = 25 \)

Step4: Calculate \( m\angle 1 \)

Substitute \( x = 25 \) into \( m\angle 1=(2x + 3)^{\circ} \)

\( m\angle 1=2\times25+3 = 53^{\circ} \)

Answer:

\( 53 \)