Question

2 - 76. 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 + 13°, 3x + 7° hint (b) c. 3x + 5°, 2x + 18°, 2x + 17° hint (c) what is the sum of the angles in a triangle? d. x, 30° hint (d)

Explanation:

Step1: Identify angle - relationship for part a

The two angles are complementary, so their sum is 90°.
$6x+(4x + 10)=90$

Step2: Simplify the equation for part a

Combine like - terms: $6x+4x+10 = 90$, which gives $10x+10 = 90$.

Step3: Solve for x in part a

Subtract 10 from both sides: $10x=90 - 10=80$. Then divide both sides by 10, so $x = 8$.

Step4: Identify angle - relationship for part b

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

Step5: Simplify the equation for part b

Combine like - terms: $5x+3x+13 + 7=180$, which gives $8x+20 = 180$.

Step6: Solve for x in part b

Subtract 20 from both sides: $8x=180 - 20 = 160$. Then divide both sides by 8, so $x = 20$.

Step7: Identify angle - relationship for part c

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

Step8: Simplify the equation for part c

Combine like - terms: $3x+2x+2x+5 + 18+17=180$, which gives $7x+40 = 180$.

Step9: Solve for x in part c

Subtract 40 from both sides: $7x=180 - 40=140$. Then divide both sides by 7, so $x = 20$.

Step10: Identify angle - relationship for part d

The sum of the interior angles of a right - triangle (one angle is 90°) is 180°.
$x+90 + 30=180$

Step11: Solve for x in part d

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

Answer:

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