18. If \( m L M=(8 x-56)^{\circ} \) and \( m H P=(5 x+22)^{\prime} \). find \( m L P \) \[ m \sqrt{P}= \] \( \qquad \) Use the circle below for questions 19-21: 19. Find the mea

To find \( m L P \), we need to equate the two given angles because they share a common vertex and are both subtended by the same arc. So, we can set them equal to each other: \[ 8x - 56 = 5x + 22. \] Now, solve for \( x \): \[ 8x - 5x = 22 + 56, \] \[ 3x = 78, \] \[ x = 26. \] Now that we have \( x \), we can substitute it back into either expression to find \( m L P \): \[ m L M = 8(26) - 56 = 208 - 56 = 152^\circ. \] So, \( m L P = 152^\circ \). Next, let’s look at some interesting tidbits! Did you know that circles have been studied for thousands of years? Ancient civilizations, like the Babylonians and Greeks, used various properties of circles in astronomy, art, and even architecture. The study of angles formed by arcs and inscribed shapes laid the groundwork for what we now know as geometry! And speaking of circles, understanding angle relationships can be a game-changer in geometry problems! A common mistake is assuming that angles that look equal are indeed equal; don’t forget to check for the subtended arcs or external angles. So, always double-check your angle relationships by looking for those hidden connections created by the circle!