Answer:
1. 5,040°
2. 168°
3. 18
4. m∠A=80°, m∠C=140°
5. a. x = 35, y = 90
b. x = 5, y = 6
Step-by-step explanation:
1. The sum of the measures of all interior angles of n-sided polygon is always equal to
[tex](n-2)\cdot 180^{\circ}[/tex]
Hence, the sum of the measures of 30-gon is
[tex](30-2)\cdot 180^{\circ}\\ \\=28\cdot 180^{\circ}\\ \\=5,040^{\circ}[/tex]
2. The measure of the interior angle in regular n-sided polygon is always equal to
[tex]\dfrac{(n-2)\cdot 180^{\circ}}{n}[/tex]
Hence, the measure of the interior angle in regular 30-gon is equal to
[tex]\dfrac{(30-2)\cdot 180^{\circ}}{30}\\ \\=\dfrac{5,040^{\circ}}{30}\\ \\=168^{\circ}[/tex]
3. The measure of the interior angle in regular n-sided polygon is always equal to
[tex]\dfrac{(n-2)\cdot 180^{\circ}}{n}[/tex]
If the measure of one interior angle in regular n-sided polygon is 160°, then
[tex]\dfrac{(n-2)\cdot 180^{\circ}}{n}=160^{\circ}\\ \\180(n-2)=160n\\ \\180n-360=160n\\ \\180n-160n=360\\ \\20n=360\\ \\n=\dfrac{360}{20}=18[/tex]
4. The diagram shows hexagon with four congruent angles with measure of [tex]7x^{\circ}[/tex] and two congruent angles with measures of [tex]4x^{\circ}.[/tex] The sum of the measures of all interior angles in a hexagon is
[tex](6-2)\cdot 180^{\circ}\\ \\=4\cdot 180^{\circ}\\ \\=720^{\circ}[/tex]
Hence,
[tex]4\cdot 7x+2\cdot 4x=720\\ \\28x+8x=720\\ \\36x=720\\ \\x=\dfrac{720}{36}=20[/tex]
Therefore,
[tex]m\angle A=(4\cdot 20)^{\circ}=80^{\circ}\\ \\m\angle C=(7\cdot 20)^{\circ}=140^{\circ}[/tex]
5. Opposite angles of the parallelogram are congruent, so
[tex]y^{\circ }=(2y-90)^{\circ}\\ \\y=2y-90\\ \\y-2y=-90\\ \\-y=-90\\ \\y=90[/tex]
and
[tex](2x+20)^{\circ}=(4x-50)^{\circ}\\ \\2x+20=4x-50\\ \\2x-4x=-50-20\\ \\-2x=-70\\ \\2x=70\\ \\x=35[/tex]
Opposite sides of the parallelogram are congruent, so
[tex]6x-15=x+10\\ \\6x-x=10+15\\ \\5x=25\\ \\x=5[/tex]
and
[tex]4y-20=2y-8\\ \\4y-2y=-8+20\\ \\2y=12\\ \\y=6[/tex]
