1. **Problem Statement:** Find the measures of the given angles and segments in triangles ΔABC and ΔJKL where G and P are incenters respectively. --- ### For ΔABC with incenter G: 2. **Recall:** The incenter is the intersection of angle bisectors and is equidistant from all sides. 3. Given angles at B and C are 20° and 11° respectively. 4. Since G is the incenter, it bisects angles at vertices. 5. Calculate m∠BCA (angle at C): Given as 11°. 6. Calculate m∠BAC (angle at A): Sum of angles in triangle is 180°, so $$m\angle BAC = 180^\circ - 20^\circ - 11^\circ = 149^\circ$$ 7. m∠ABG is half of m∠ABC (since G bisects angle B): $$m\angle ABC = 20^\circ \Rightarrow m\angle ABG = \frac{20^\circ}{2} = 10^\circ$$ 8. m∠BAG is half of m∠BAC: $$m\angle BAC = 149^\circ \Rightarrow m\angle BAG = \frac{149^\circ}{2} = 74.5^\circ$$ 9. DG, BE, BG, GC are segments related to the incenter and sides. Since G is incenter, DG and BE are perpendicular distances from G to sides. 10. Using given side lengths and right angles, DG = 4 (given), BE = 11 (given). 11. BG and GC are segments from vertices to incenter G. Since G lies on angle bisectors, BG and GC can be found using triangle properties or given data. 12. BG = 11 (given), GC = 4 (given). --- ### For ΔJKL with incenter P: 13. Given angles at J: (7x - 6)° and (5x + 4)°, and at L: 26°. 14. Sum of angles in triangle: $$ (7x - 6) + (5x + 4) + 26 = 180 $$ 15. Simplify: $$ 7x - 6 + 5x + 4 + 26 = 180 $$ $$ 12x + 24 = 180 $$ $$ 12x = 156 $$ $$ x = 13 $$ 16. Find m∠JKP (angle at J bisected by P): $$ m\angle JKP = \frac{(7x - 6)^\circ}{2} = \frac{7(13) - 6}{2} = \frac{91 - 6}{2} = \frac{85}{2} = 42.5^\circ $$ 17. Find MP using given expressions 3x + 14 and 9x - 34: $$ 3x + 14 = 3(13) + 14 = 39 + 14 = 53 $$ $$ 9x - 34 = 9(13) - 34 = 117 - 34 = 83 $$ Since MP is likely the segment between these points, if equal, set equal: $$ 3x + 14 = 9x - 34 $$ $$ 14 + 34 = 9x - 3x $$ $$ 48 = 6x $$ $$ x = 8 $$ Recalculate MP: $$ 3(8) + 14 = 24 + 14 = 38 $$ $$ 9(8) - 34 = 72 - 34 = 38 $$ So, MP = 38. 18. Find PJ using 22, 2x + 3, 5x - 45: Sum of segments around P: $$ 22 + (2x + 3) + (5x - 45) = 0 $$ Or if PJ is one of these, solve for x: Set equal 2x + 3 = 5x - 45: $$ 2x + 3 = 5x - 45 $$ $$ 3 + 45 = 5x - 2x $$ $$ 48 = 3x $$ $$ x = 16 $$ Calculate PJ: $$ 2(16) + 3 = 32 + 3 = 35 $$ $$ 5(16) - 45 = 80 - 45 = 35 $$ PJ = 35. --- **Final answers:** 10. $m\angle ABG = 10^\circ$ 11. $m\angle BCA = 11^\circ$ 12. $m\angle BAC = 149^\circ$ 13. $m\angle BAG = 74.5^\circ$ 14. $DG = 4$ 15. $BE = 11$ 16. $BG = 11$ 17. $GC = 4$ 18. $m\angle JKP = 42.5^\circ$ 19. $MP = 38$ 20. $PJ = 35$