Question

structure find the measure of each angle in the diagram. (3y + 11)° (4x - 22)° (7x + 4)° 10y° (3y + 11)°: ° 10y°: ° (4x - 22)°: ° (7x + 4)°: °

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $3y + 11=10y$.
Solve for $y$:
$11 = 10y-3y$
$11 = 7y$
$y=\frac{11}{7}$

Step2: Use linear - pair property

$(4x - 22)+(7x + 4)=180$ (since they are a linear - pair of angles and linear - pair angles are supplementary).
Combine like terms: $4x+7x-22 + 4=180$
$11x-18 = 180$
$11x=180 + 18$
$11x=198$
$x = 18$

Step3: Calculate the angle measures

For $(3y + 11)^{\circ}$:
Substitute $y=\frac{11}{7}$ into $3y + 11$:
$3\times\frac{11}{7}+11=\frac{33}{7}+11=\frac{33 + 77}{7}=\frac{110}{7}\approx15.71^{\circ}$
For $10y^{\circ}$:
Substitute $y=\frac{11}{7}$ into $10y$:
$10\times\frac{11}{7}=\frac{110}{7}\approx15.71^{\circ}$
For $(4x - 22)^{\circ}$:
Substitute $x = 18$ into $4x-22$:
$4\times18-22=72-22 = 50^{\circ}$
For $(7x + 4)^{\circ}$:
Substitute $x = 18$ into $7x + 4$:
$7\times18+4=126 + 4=130^{\circ}$

Answer:

$(3y + 11)^{\circ}:\frac{110}{7}\approx15.71$
$10y^{\circ}:\frac{110}{7}\approx15.71$
$(4x - 22)^{\circ}:50$
$(7x + 4)^{\circ}:130$