Question

2 - 76. for each diagram below, solve for x. explain what relationship(s) from your angle relationships toolkit you used for each problem

Explanation:

Step1: Identify angle - sum property for part a

The two angles are complementary (sum to 90°). So, $6x+(4x + 10)=90$.

Step2: Simplify the equation for part a

Combine like - terms: $6x+4x+10 = 90$, which gives $10x+10 = 90$. Then subtract 10 from both sides: $10x=90 - 10=80$. Divide both sides by 10: $x = 8$.

Step3: Identify angle - sum property for part b

The two angles are supplementary (sum to 180°). So, $(5x + 13)+(3x+7)=180$.

Step4: Simplify the equation for part b

Combine like - terms: $5x+3x+13 + 7=180$, which gives $8x+20 = 180$. Subtract 20 from both sides: $8x=180 - 20 = 160$. Divide both sides by 8: $x = 20$.

Step5: Identify angle - sum property for part c

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

Step6: Simplify the equation for part c

Combine like - terms: $3x+2x+2x+5 + 18+17 = 180$, which gives $7x+40 = 180$. Subtract 40 from both sides: $7x=180 - 40 = 140$. Divide both sides by 7: $x = 20$.

Step7: Identify angle - sum property for part d

The sum of the interior angles of a right - triangle is 180°, and one angle is 90° and another is 30°. So, $x+90 + 30=180$.

Step8: Simplify the equation for part d

Combine like - terms: $x+120 = 180$. Subtract 120 from both sides: $x=180 - 120 = 60$.

Answer:

a. $x = 8$
b. $x = 20$
c. $x = 20$
d. $x = 60$