Ingles \( A \) and \( B \) are corresponding angles formed by two parallel lines cut by a transversal. If \( n \angle A=4 x \) and \( m \angle B=3 x+7 \), find the value of \( x \). Explain. \[ 4 x=3 x+7 \] ingles \( G \) and \( H \) are supplementary and congruent. If \( \angle G \) and \( \angle H \) are alternate interior angle

To solve the problem, we need to analyze the information given about angles \( A \) and \( B \), which are corresponding angles formed by two parallel lines cut by a transversal. ### Step 1: Set up the equation We know that corresponding angles are equal when two parallel lines are cut by a transversal. Therefore, we can set up the equation based on the information provided: \[ 4x = 3x + 7 \] ### Step 2: Solve for \( x \) To find the value of \( x \), we will isolate \( x \) in the equation. 1. Subtract \( 3x \) from both sides: \[ 4x - 3x = 7 \] This simplifies to: \[ x = 7 \] ### Step 3: Verify the solution Now, let's verify the values of angles \( A \) and \( B \) using \( x = 7 \): - Calculate \( m \angle A \): \[ m \angle A = 4x = 4(7) = 28 \] - Calculate \( m \angle B \): \[ m \angle B = 3x + 7 = 3(7) + 7 = 21 + 7 = 28 \] Since both angles \( A \) and \( B \) are equal (28 degrees), our solution is verified. ### Step 4: Analyze angles \( G \) and \( H \) Next, we consider angles \( G \) and \( H \), which are supplementary and congruent. - Since they are congruent, we can denote: \[ m \angle G = m \angle H = y \] - Since they are supplementary, we have: \[ m \angle G + m \angle H = 180 \] Substituting the values: \[ y + y = 180 \] This simplifies to: \[ 2y = 180 \] Dividing both sides by 2 gives: \[ y = 90 \] Thus, both angles \( G \) and \( H \) are \( 90 \) degrees, confirming they are congruent and supplementary. ### Final Answers - The value of \( x \) is \( 7 \). - Angles \( G \) and \( H \) are both \( 90 \) degrees.