When a sheet of paper is folded, the angle formed by the fold line with the original edge is equal to the angle formed by the fold line with the folded edge. In this case, the angle 1 is formed by the original edge and the fold line. The angle 3 (which is not explicitly labeled but is the reflection of 1 across the fold line) will be equal to 1.

When a sheet of paper is folded, the angle formed by the fold line with the original edge is equal to the angle formed by the fold line with the folded edge. In this case, the angle 1 is formed by the original edge and the fold line. The angle 3 (which is not explicitly labeled but is the reflection of 1 across the fold line) will be equal to 1.

From the previous steps, we know that 1 = 3 and 4 = 2. Substituting these into the equation for angles on a straight line (2 1 + 4 = 180^°), we get 2 1 + 2 = 180^°.

Given that 1 = 58^°, we can substitute this value into the equation 2 1 + 2 = 180^°. This gives us 2 × 58^° + 2 = 180^°. Solving for 2, we get 2 = 180^° - 116^° = 64^°.

A rectangular sheet is folded as shown and then unfolded. If $\angle 1=58^\circ$ , find $\angle 2$ .

Answer: $\angle 2 = 64^\circ$

Step-by-step solution

Step 1: Understand the folding property

When a sheet of paper is folded, the angle formed by the fold line with the original edge is equal to the angle formed by the fold line with the folded edge. In this case, the angle $\angle 1$ is formed by the original edge and the fold line. The angle $\angle 3$ (which is not explicitly labeled but is the reflection of $\angle 1$ across the fold line) will be equal to $\angle 1$ .

Step 2: Identify angles on a straight line

The angles $\angle 1$ , $\angle 3$ , and $\angle 4$ (where $\angle 4$ is the angle between the fold line and the top edge of the folded part) together form a straight angle on the bottom edge of the rectangle. Therefore, their sum is $180^\circ$ . Since $\angle 1 = \angle 3$ , we can write this as $2\angle 1 + \angle 4 = 180^\circ$ .

Step 3: Relate angles to parallel lines

Since the sheet is rectangular, its opposite sides are parallel. The fold line acts as a transversal intersecting these parallel lines. The angle $\angle 4$ (the angle between the fold line and the top edge of the folded part) and $\angle 2$ (the angle between the fold line and the top edge of the unfolded part) are alternate interior angles. Therefore, they are equal.

Step 4: Substitute and solve for $\angle 2$

From the previous steps, we know that $\angle 1 = \angle 3$ and $\angle 4 = \angle 2$ . Substituting these into the equation for angles on a straight line ( $2\angle 1 + \angle 4 = 180^\circ$ ), we get $2\angle 1 + \angle 2 = 180^\circ$ .

Step 5: Calculate the value of $\angle 2$

Given that $\angle 1 = 58^\circ$ , we can substitute this value into the equation $2\angle 1 + \angle 2 = 180^\circ$ . This gives us $2 \times 58^\circ + \angle 2 = 180^\circ$ . Solving for $\angle 2$ , we get $\angle 2 = 180^\circ - 116^\circ = 64^\circ$ .

A rectangular sheet is folded as shown and then unfolded. If ∠1=58∘\angle 1=58^\circ∠1=58∘, find ∠2\angle 2∠2.