Answer:
<h2>Q1. y = 3</h2><h2>Q2. m∠B = 125°</h2><h2>Q3. m∠SPQ = 110°</h2><h2>Q4. x = 55</h2><h2>Q5. x = 123</h2>
Step-by-step explanation:
<h3>Q1.</h3>
If ABCD is a rectangle, then diagonals are congruent: AC = BD.
We have
AC = 5y - 2
BD = 4y + 1
Substitute:
5y - 2 = 4y + 1 <em>add 2 to both sides</em>
5y - 2 + 2 = 4y + 1 + 2
5y = 4y + 3 <em>subtract 4y from both sides</em>
5y - 4y = 4y - 4y + 3
y = 3
<h3>
Q2.</h3>
In a parallelogram opposite angles are congruent. Therefore m∠D = m∠B.
We have m∠D = 125° → m∠B = 125°
<h3>Q3.</h3>
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 <em>subtract 15 from both sides</em>
2x + 15 - 15 = 3x - 5 - 15
2x = 3x - 20 <em>subtract 3x from both sides</em>
2x - 3x = 3x - 3x - 20
-x = -20 <em>change the signs</em>
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°
<h3>Q4.</h3>
In the parallelogram, the sum of the angle measures on one side is 180°.
Therefore we have the equation:
(2x + 15) + x = 180 <em>combine like terms</em>
(2x + x) + 15 = 180 <em>subtract 15 from both sides</em>
3x + 15 - 15 = 180 - 15
3x = 165 <em>divide both sides by 3</em>
3x/3 = 165/3
x = 55
<h3>Q5.</h3>
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 <em>combine like terms</em>
(2y + y) + 9 = 180 <em>subtract 9 from both sides</em>
3y + 9 - 9 = 180 - 9
3y = 171 <em>divide both sides by 3</em>
y = 57
Put it to (*):
x = 2(57) + 9
x = 114 + 9
x = 123
