Question

2 - 84. review: solve for x using the theorems and properties of angles.

Explanation:

Step1: Identify angle - relationship in part a

In a right - triangle, the sum of the two non - right angles is 90°. So, we set up the equation $(5x + 8)+(3x - 4)=90$.

Step2: Simplify the left - hand side of the equation in part a

Combine like terms: $5x+3x + 8 - 4=90$, which gives $8x+4 = 90$.

Step3: Solve for x in part a

Subtract 4 from both sides: $8x=90 - 4=86$. Then divide both sides by 8: $x=\frac{86}{8}=\frac{43}{4}=10.75$.

Step4: Identify angle - relationship in part b

Vertical angles are equal. So, we set up the equation $16x+4 = 80$.

Step5: Solve for x in part b

Subtract 4 from both sides: $16x=80 - 4 = 76$. Then divide both sides by 16: $x=\frac{76}{16}=\frac{19}{4}=4.75$.

Step6: Identify angle - relationship in part c

Vertical angles are equal. So, $x=135$.

Step7: Identify angle - relationship in part d

Corresponding angles are equal. So, we set up the equation $x + 146=135$.

Step8: Solve for x in part d

Subtract 146 from both sides: $x=135 - 146=-11$.

Answer:

a. $x = 10.75$
b. $x = 4.75$
c. $x = 135$
d. $x=-11$