Question

1. construct arguments write a proof. given: m∠tuv = 90 prove: x = 12
2. construct arguments write an indirect proof by proving the contrapositive. given: gj = 48 prove: x ≠ 12

Explanation:

46.

Step1: Use vertical - angle property

Vertical angles are equal. So, the angle with measure $y^{\circ}$ in the left - hand figure is equal to the angle with measure $4x^{\circ}$. In the right - hand figure, since $\angle TUV = 90^{\circ}$, we know that $y+42 = 90$.

Step2: Solve for $y$

Subtract 42 from both sides of the equation $y + 42=90$. So, $y=90 - 42=48$.

Step3: Equate and solve for $x$

Since $y = 4x$ and $y = 48$, we set up the equation $4x=48$. Divide both sides by 4: $x=\frac{48}{4}=12$.

Step1: State the contrapositive

The contrapositive of the statement "If $GJ = 48$, then $x
eq12$" is "If $x = 12$, then $GJ
eq48$".

Step2: Calculate $GJ$ when $x = 12$

Given that $GJ=2x + x=3x$. If $x = 12$, then $GJ=3\times12 = 36$.

Step3: Conclude the proof

Since when $x = 12$, $GJ = 36
eq48$, the contrapositive is true. Since the contrapositive is true, the original statement "If $GJ = 48$, then $x
eq12$" is also true.

Answer:

$x = 12$

47.