Need with questions 12 to 15 both a's and b's. show work step by step and theorems that justifies/supports the respective answers.

Step-by-step explanation:

11. If a quadrilateral has 2 sets of opposite angles congruent, then it is a parallelogram.

∠J ≅ ∠ L (congruent ∠s have equal measures.)

9y+1 = 10y-13

when y = 14

9(14)+1 = 10(14)-13

126+1 = 140-13

127 ≅ 127  (congruent ∠s have equal measures.)

          or

THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram.

∠K + ∠ L = 180° ( angles are supplementary )

(10y-13) + (7x+4) = 180

10y-13+7x+4 = 180°

y=14   x =7

10(14) -13 + 7(7) + 4 = 180

140 - 13 + 49 + 4 = 180

rearrange the plus sign

140+49+4-13=180

193-13=180

180≅180

9

12.  

If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.

IADI = IBCI

IDCI = IABI

therefore IACI = IBDI = 25

corres. parts of congruent triangles are congruent.

13. If a parallelogram has perpendicular diagonals, it is a rhombus.

∠PQR=∠PSR= 2x+3

All right angles are congruent.