The interior angles of the hexagon are 2x°, 1/2 x°,x + 40°, 110°, 130°, 160°. Find the value of the smallest angle. [3 Marks]
The Correct Answer and Explanation is:
To solve for the smallest angle in the hexagon, we need to use the fact that the sum of the interior angles of any polygon is calculated by the formula:Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n – 2) \times 180^\circSum of interior angles=(n−2)×180∘
where nnn is the number of sides of the polygon. Since we are dealing with a hexagon, n=6n = 6n=6. So, the sum of the interior angles of a hexagon is:(6−2)×180∘=4×180∘=720∘(6 – 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ(6−2)×180∘=4×180∘=720∘
Given that the interior angles of the hexagon are 2x∘2x^\circ2x∘, 12x∘\frac{1}{2}x^\circ21x∘, x+40∘x + 40^\circx+40∘, 110∘110^\circ110∘, 130∘130^\circ130∘, and 160∘160^\circ160∘, we can set up an equation to find the value of xxx:2x+12x+(x+40)+110+130+160=7202x + \frac{1}{2}x + (x + 40) + 110 + 130 + 160 = 7202x+21x+(x+40)+110+130+160=720
Now, let’s simplify this equation:2x+12x+x+40+110+130+160=7202x + \frac{1}{2}x + x + 40 + 110 + 130 + 160 = 7202x+21x+x+40+110+130+160=720
Combine like terms:(2x+12x+x)+(40+110+130+160)=720(2x + \frac{1}{2}x + x) + (40 + 110 + 130 + 160) = 720(2x+21x+x)+(40+110+130+160)=720 (3x+12x)+440=720\left( 3x + \frac{1}{2}x \right) + 440 = 720(3x+21x)+440=720
To combine the terms with xxx, we convert 3x3x3x to a fraction:6×2+1×2=7×2\frac{6x}{2} + \frac{1x}{2} = \frac{7x}{2}26x+21x=27x
So the equation becomes:7×2+440=720\frac{7x}{2} + 440 = 72027x+440=720
Now, subtract 440 from both sides:7×2=280\frac{7x}{2} = 28027x=280
Multiply both sides by 2:7x=5607x = 5607x=560
Now, divide both sides by 7:x=80x = 80x=80
Now that we have x=80∘x = 80^\circx=80∘, we can substitute it back into the expressions for the angles:
- The first angle is 2x=2(80)=160∘2x = 2(80) = 160^\circ2x=2(80)=160∘,
- The second angle is 12x=12(80)=40∘\frac{1}{2}x = \frac{1}{2}(80) = 40^\circ21x=21(80)=40∘,
- The third angle is x+40=80+40=120∘x + 40 = 80 + 40 = 120^\circx+40=80+40=120∘,
- The other angles are given as 110°, 130°, and 160°.
Thus, the smallest angle is 40∘40^\circ40∘.
Conclusion:
The smallest angle in the hexagon is 40∘40^\circ40∘.