Answer:
3) [tex]\angle 7 = 61^\circ[/tex]
4) [tex]61 + (2x - 5) = 180[/tex]
5) [tex] x = 62 [/tex]
6) 426 ft²
Step-by-step explanation:
3)
Vertically Opposite Angles are formed opposite each other when two lines intersect. They are always equal.
- Given: [tex]\angle 7 = \angle 5[/tex], and [tex]\angle 5 = 61^\circ[/tex].
By the property of vertically opposite angles:
[tex]\angle 7 = \angle 5 = 61^\circ[/tex]
[tex]\dotfill [/tex]
4) [tex] \angle 6 = 2x - 5 [/tex]
Here:
[tex]\angle 6[/tex] and [tex]\angle 5[/tex] form a linear pair.
Linear pairs are adjacent angles whose sum is [tex]180^\circ[/tex].
Since [tex]\angle 5 = 61^\circ[/tex], and [tex]\angle 6 = 2x - 5[/tex], we can set up the equation as:
[tex]\angle 5 + \angle 6 = 180^\circ[/tex].
Therefore, the equation is:
[tex]61 + (2x - 5) = 180[/tex]
[tex]\dotfill[/tex]
5)
From the equation [tex]61^\circ + (2x - 5) = 180^\circ[/tex], solve for [tex]x[/tex].
[tex] 61^\circ + (2x - 5) = 180^\circ [/tex]
[tex] 2x + 56 = 180 [/tex]
[tex] 2x = 180 - 56 [/tex]
[tex] 2x = 124 [/tex]
[tex] x = \dfrac{124}{2} [/tex]
[tex] x = 62 [/tex]
[tex]\dotfill[/tex]
6) Surface Area of Rectangular Prism:
Given:
- Length [tex]l = 12[/tex] ft,
- Width [tex]w = 9[/tex] ft,
- Height [tex]h = 5[/tex] ft.
Formula:
Surface Area:
[tex]S = 2lw + 2wh + 2lh[/tex]
Substitute the value and simplify:
[tex] S = 2(12 \times 9) + 2(9 \times 5) + 2(12 \times 5) [/tex]
[tex] S = 2(108) + 2(45) + 2(60) [/tex]
[tex] S = 216 + 90 + 120 [/tex]
[tex] S = 426 \, \text{ft}^2 [/tex]
So, the surface area of the rectangular prism is 426 ft².
