Question

quiz
interior and exterior angles of triangles and polygons
complete:

1. ( mangle a = )
2. ( mangle cbd = )
3. ( mangle abc = )

complete:

1. ( mangle 4 = mangle 2 + mangle )
2. if ( mangle 1 = 55^{circ} ) and ( mangle 2 = 60^{circ} ), then ( mangle 3 = )
3. if ( mangle 5 = 128^{circ} ), ( mangle 1 = x ) and ( mangle 2 = x + 10 ), then ( x = )
4. find values of ( x,y,z )
5. find values of ( x ) and ( y )
6. is it possible to create a triangle out of sides 2,4,6?
7. the measures of two sides are given. between what two numbers must the third side fall,

write your answer as inequality.
4,10

1. in a regular hexagon, the sum of measures of the exterior

angles is __ and the measure of each interior angle is __

1. if the measure of each angle of a regular polygon is ( 170^{circ} ), then

the measure of each exterior angle is ____ and the polygon has
____ sides.

Explanation:

7) Find values of \(x,y,z\)

Step1: Find \(x\)

Use the property that the sum of angles on a straight - line is \(180^{\circ}\).
\(x + 115^{\circ}=180^{\circ}\)
\(x=180^{\circ}-115^{\circ}=65^{\circ}\)

Step2: Find \(y\)

Use the angle - sum property of a triangle (\(180^{\circ}\)).
In the triangle with angles \(y\), \(60^{\circ}\), and \(x = 65^{\circ}\), we have \(y+60^{\circ}+x = 180^{\circ}\)
Substitute \(x = 65^{\circ}\)
\(y+60^{\circ}+65^{\circ}=180^{\circ}\)
\(y=180^{\circ}-(60^{\circ}+65^{\circ})=55^{\circ}\)

Step3: Find \(z\)

Use the angle - sum property of a right - triangle (\(90^{\circ}+z + 115^{\circ}\)’s supplementary angle \(=180^{\circ}\)).
The supplementary angle of \(115^{\circ}\) is \(65^{\circ}\) (from Step1). In the right - triangle (\(90^{\circ}\) angle), \(90^{\circ}+z+65^{\circ}=180^{\circ}\)
\(z=180^{\circ}-(90^{\circ}+65^{\circ}) = 25^{\circ}\)

8) Find values of \(x\) and \(y\)

Step1: Find \(y\)

Use the property of vertical angles and the fact that the sum of angles in a right - triangle is \(180^{\circ}\).
\(y + 60^{\circ}=90^{\circ}\) (complementary angles in a right - triangle)
\(y = 30^{\circ}\)

Step2: Find \(x\)

Use the property of parallel lines (alternate interior angles).
\(x=y\) (alternate interior angles)
\(x = 30^{\circ}\)

9) Check if sides \(2\), \(4\), \(6\) form a triangle

Step1: Apply the triangle - inequality theorem

The triangle - inequality theorem states that for three sides \(a\), \(b\), \(c\) of a triangle, \(a + b>c\), \(a + c>b\), and \(b + c>a\)
Let \(a = 2\), \(b = 4\), \(c = 6\)
\(a + b=2 + 4=6
ot>6\)

10) Find the range of the third side

Step1: Apply the triangle - inequality formula

If two sides of a triangle are \(a = 4\) and \(b = 10\), and the third side is \(c\)
\(\vert a - b\vert\(\vert4 - 10\vert\(6

11) Sum of exterior angles of a regular hexagon and measure of each interior angle

Step1: Sum of exterior angles

The sum of exterior angles of any polygon is \(360^{\circ}\)

Step2: Measure of each interior angle

The formula for the measure of an interior angle of a regular polygon is \(\frac{(n - 2)\times180^{\circ}}{n}\), where \(n = 6\) (for a hexagon)
\(\frac{(6 - 2)\times180^{\circ}}{6}=\frac{4\times180^{\circ}}{6}=120^{\circ}\)

12) Measure of each exterior angle and number of sides of a regular polygon

Step1: Measure of each exterior angle

Since the sum of an interior angle (\(I\)) and an exterior angle (\(E\)) of a polygon is \(180^{\circ}\), if \(I = 170^{\circ}\), then \(E=180^{\circ}-170^{\circ}=10^{\circ}\)

Step2: Number of sides

The formula for the measure of an exterior angle of a regular polygon is \(E=\frac{360^{\circ}}{n}\)
If \(E = 10^{\circ}\), then \(n=\frac{360^{\circ}}{10^{\circ}}=36\)

Answer:

1. \(x = 65^{\circ}\), \(y = 55^{\circ}\), \(z = 25^{\circ}\)
2. \(x = 30^{\circ}\), \(y = 30^{\circ}\)
3. No
4. \(6 < c<14\)
5. Sum of exterior angles \(=360^{\circ}\), Measure of each interior angle \(=120^{\circ}\)
6. Measure of each exterior angle \(=10^{\circ}\), Number of sides \(=36\)