Question

(11y - 36)° and 63° are vertical angles; (4x + 45)° and (5x - 18)° have a relationship; (4n + 22)° and (8n - 10)° are adjacent angles on a straight line; (9y + 7)° and (2y + 98)° are vertical angles. for each, find the relationship, equation, and solve for the variable.

Explanation:

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Top Left (Solve for y):

Step1: Identify vertical angles

Vertical angles are congruent, so $11y - 36 = 63$.

Step2: Isolate variable term

Add 36 to both sides: $11y = 63 + 36$
$11y = 99$

Step3: Solve for y

Divide by 11: $y = \frac{99}{11}$
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Top Right (Solve for x):

Step1: Identify supplementary angles

Adjacent angles on a line sum to $180^\circ$, so $5x - 18 + 4x + 45 = 180$.

Step2: Simplify left side

Combine like terms: $9x + 27 = 180$

Step3: Isolate variable term

Subtract 27: $9x = 180 - 27$
$9x = 153$

Step4: Solve for x

Divide by 9: $x = \frac{153}{9}$
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Bottom Left (Solve for n):

Step1: Identify supplementary angles

Angles form a linear pair, sum to $180^\circ$, so $4n + 22 + 8n - 10 = 180$.

Step2: Simplify left side

Combine like terms: $12n + 12 = 180$

Step3: Isolate variable term

Subtract 12: $12n = 180 - 12$
$12n = 168$

Step4: Solve for n

Divide by 12: $n = \frac{168}{12}$
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Bottom Right (Solve for y):

Step1: Identify vertical angles

Vertical angles are congruent, so $9y + 7 = 2y + 98$.

Step2: Isolate variable terms

Subtract $2y$: $7y + 7 = 98$

Step3: Isolate variable term

Subtract 7: $7y = 98 - 7$
$7y = 91$

Step4: Solve for y

Divide by 7: $y = \frac{91}{7}$
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Answer:

Top Left:

Relationship: Vertical Angles
Equation: $11y - 36 = 63$
$y = 9$

Top Right:

Relationship: Supplementary Angles
Equation: $5x - 18 + 4x + 45 = 180$
$x = 17$

Bottom Left:

Relationship: Supplementary Angles (Linear Pair)
Equation: $4n + 22 + 8n - 10 = 180$
$n = 14$

Bottom Right:

Relationship: Vertical Angles
Equation: $9y + 7 = 2y + 98$
$y = 13$