Question

1. for each diagram below, solve for x. explain what relationship(s) from your angle relationships toolkit you used for each problem. a. 6x 4x + 10° hint (a): b. 5x + 12° 3x + 7° hint (b): c. 2x + 18° 3x + 5° 2x + 17° hint (c): d. 30° x hint (d):

Explanation:

a.

Step1: Use vertical - angle property

Vertical angles are equal. So, $6x=4x + 10$.

Step2: Solve the equation for $x$

Subtract $4x$ from both sides: $6x-4x=4x + 10-4x$. This gives $2x=10$. Then divide both sides by 2: $x = 5$.

Step1: Use the linear - pair property

The sum of angles in a linear - pair is $180^{\circ}$. So, $(5x + 13)+(3x+7)=180$.

Step2: Combine like terms

$5x+3x+13 + 7=180$, which simplifies to $8x+20 = 180$.

Step3: Isolate the variable term

Subtract 20 from both sides: $8x+20-20=180 - 20$, getting $8x=160$.

Step4: Solve for $x$

Divide both sides by 8: $x = 20$.

Step1: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, $(2x + 17)+(2x+18)+(3x + 5)=180$.

Step2: Combine like terms

$2x+2x+3x+17 + 18+5=180$, which simplifies to $7x+40 = 180$.

Step3: Isolate the variable term

Subtract 40 from both sides: $7x+40-40=180 - 40$, getting $7x=140$.

Step4: Solve for $x$

Divide both sides by 7: $x = 20$.

Answer:

$x = 5$

b.