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Step-by-step explanation:

If ABCD is a rectangle, then diagonals are congruent: AC = BD.

We have

AC = 5y - 2

BD = 4y + 1

Substitute:

5y - 2 = 4y + 1           add 2 to both sides

5y - 2 + 2 = 4y + 1 + 2

5y = 4y + 3            subtract 4y from both sides

5y - 4y = 4y - 4y + 3

y = 3

In a parallelogram opposite angles are congruent. Therefore m∠D = m∠B.

We have m∠D = 125° → m∠B = 125°

In the rhombus, the diagonals are bisectors of the rhombus angles.

Therefore ∠SPR and ∠QPR are congruent.

We have

m∠SPR = (2x +15)°

m∠QPR = (3x - 5)°

The equation:

2x + 15 = 3x - 5           subtract 15 from both sides

2x + 15 - 15 = 3x - 5 - 15

2x = 3x - 20         subtract 3x from both sides

2x - 3x = 3x - 3x - 20

-x = -20           change the signs

x = 20

Substitute it to the expression m∠SPR = (2x + 15)°:

m∠SPR = (2(20) + 15)° = (40 + 15)° = 55°

m∠SPR = m∠QPR → m∠QPR = 55°

∠SPQ = ∠SPR + ∠QPR → m∠SPQ = 2(55°) = 110°

In the parallelogram, the sum of the angle measures on one side is 180°.

Therefore we have the equation:

(2x + 15) + x = 180      combine like terms

(2x + x) + 15 = 180           subtract 15 from both sides

3x + 15 - 15 = 180 - 15

3x = 165            divide both sides by 3

3x/3 = 165/3

x = 55

In a parallelogram opposite angles are congruent.

Therefore z = y and x = 2z + 9 → x = 2y + 9    (*)

In the parallelogram, the sum of the angle measures on one side is 180°.

Therefore x + y = 180     (**)

Substitute (*) to (**)

(2y + 9) + y = 180          combine like terms

(2y + y) + 9 = 180             subtract 9 from both sides

3y + 9 - 9 = 180 - 9

3y = 171          divide both sides by 3

y = 57

Put it to (*):

x = 2(57) + 9

x = 114 + 9

x = 123

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step-by-step explanation:

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