Direction: In parallelogram HOPE, find the value of \( x \) and the indicated angles. \[ \xrightarrow{\text { Direction: In parallelogram }} \] 1. \( x \) 2. \( \mathrm{m} \angle O H E \) 3. \( \mathrm{m} \angle O P E \) \[ (3 x-15)^{\circ} \] 4. \( \mathrm{m} \angle H O P \) 5. \( \mathrm{m} \angle H E P \)

In parallelogram HOPE, opposite angles are equal, and consecutive angles are supplementary. To find \( x \), set up the equation using the fact that \( \mathrm{m} \angle OHE + \mathrm{m} \angle OPE = 180^\circ\). Given \( \mathrm{m} \angle OHE = (3x - 15)^\circ\) and letting \( \mathrm{m} \angle OPE = (3x - 15)^\circ \) as well, then: \[ (3x - 15) + (3x - 15) = 180 \Rightarrow 6x - 30 = 180 \Rightarrow 6x = 210 \Rightarrow x = 35. \] Now substituting \( x \) back into the equations we get: 1. \( x = 35 \) 2. \( \mathrm{m} \angle OHE = 3(35) - 15 = 105^\circ \) 3. \( \mathrm{m} \angle OPE = 75^\circ \) (since \( 180 - 105 = 75 \)) 4. Since \( \angle HOP = \angle OHE\), \( \mathrm{m} \angle HOP = 105^\circ \). 5. Lastly, \( \mathrm{m} \angle HEP = \mathrm{m} \angle OPE = 75^\circ \). So we have the values: 1. \( x = 35 \) 2. \( \mathrm{m} \angle OHE = 105^\circ \) 3. \( \mathrm{m} \angle OPE = 75^\circ \) 4. \( \mathrm{m} \angle HOP = 105^\circ \) 5. \( \mathrm{m} \angle HEP = 75^\circ \) Angles in a parallelogram dance together like a well-rehearsed team, with opposite angles mirroring each other and adjacent angles always on the same page—adding up to a perfect 180 degrees. So, if one angle is feeling a bit obtuse, you can count on its neighbor for a complementary pep talk! To dive deeper, consider the properties of parallelograms! They not only have equal opposite angles but also equal opposite sides. This means if you ever find yourself in a jam trying to solve for angles or lengths, just remember: the structure of a parallelogram is your geometric best friend, providing unwavering support!