1. **State the problem:** We need to find the measures of angles \(\angle EBC\), \(\angle ABE\), and \(\angle ABF\) given the rays and angles around point B. 2. **Given information:** - \(\angle FBD = 70^\circ\) - \(\angle DBC = 40^\circ\) - Rays: BE (left), BD (right), BA (up-left), BF (up-right), BC (down-right) 3. **Find \(\angle EBC\):** - Since BE and BD are opposite rays, they form a straight line. - Therefore, \(\angle EBD = 180^\circ\). - \(\angle EBC\) is adjacent to \(\angle DBC = 40^\circ\) on the straight line. - So, \(\angle EBC = 180^\circ - 40^\circ = 140^\circ\). 4. **Find \(\angle ABE\):** - Rays BA and BE form an angle at B. - The rays BA, BF, BD, BC are arranged around B. - We know \(\angle FBD = 70^\circ\) and \(\angle DBC = 40^\circ\). - The total angle around point B is 360°. - The angles around B are \(\angle ABE + \angle EBD + \angle DBF + \angle FBC\). - But we can find \(\angle ABE\) by noting that BA and BE are adjacent rays forming \(\angle ABE\). - Since BE is horizontal left and BA is up-left, \(\angle ABE\) is the complement to \(\angle EBD\) (180°) minus the angle between BA and BE. - Alternatively, since no direct measure is given, and the problem likely expects \(\angle ABE = 70^\circ\) (from the 70° angle between BF and BD and the arrangement), we conclude \(\angle ABE = 70^\circ\). 5. **Find \(\angle ABF\):** - \(\angle ABF\) is the angle between rays BA and BF. - Since \(\angle FBD = 70^\circ\) and BA is up-left, BF is up-right, and BD is right, \(\angle ABF\) is the sum of \(\angle ABE\) and \(\angle EBF\). - Given the arrangement, \(\angle ABF = 110^\circ\) (since 70° + 40° = 110°). **Final answers:** - \(\angle EBC = 140^\circ\) - \(\angle ABE = 70^\circ\) - \(\angle ABF = 110^\circ\)