**\#5.) Sahara says that \( -135^{\circ} \) is coterminal to \( 565^{\circ} \) since they end up in the same quadrant. Do you agree or disagree with her? Explain your reasoning and show work to support it.

To determine whether \( -135^{\circ} \) is coterminal with \( 565^{\circ} \), we need to check if they have the same terminal side when drawn in standard position. Two angles are coterminal if they differ by a multiple of \( 360^{\circ} \). ### Step 1: Find the coterminal angle of \( -135^{\circ} \) To find a positive coterminal angle for \( -135^{\circ} \), we can add \( 360^{\circ} \): \[ -135^{\circ} + 360^{\circ} = 225^{\circ} \] ### Step 2: Find the coterminal angle of \( 565^{\circ} \) Next, we can find a coterminal angle for \( 565^{\circ} \) by subtracting \( 360^{\circ} \): \[ 565^{\circ} - 360^{\circ} = 205^{\circ} \] ### Step 3: Compare the angles Now we have: - The positive coterminal angle of \( -135^{\circ} \) is \( 225^{\circ} \). - The coterminal angle of \( 565^{\circ} \) is \( 205^{\circ} \). ### Step 4: Determine if they are coterminal Since \( 225^{\circ} \) and \( 205^{\circ} \) are not equal, \( -135^{\circ} \) and \( 565^{\circ} \) are not coterminal. ### Conclusion I disagree with Sahara's statement. \( -135^{\circ} \) is not coterminal with \( 565^{\circ} \) because they do not share the same terminal side.