Answer:
Q1. y = 3
Q2. m∠B = 125°
Q3. m∠SPQ = 110°
Q4. x = 55
Q5. x = 123
Step-by-step explanation:
Q1.
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
Q2.
In a parallelogram opposite angles are congruent. Therefore m∠D = m∠B.
We have m∠D = 125° → m∠B = 125°
Q3.
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°
Q4.
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
Q5.
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
