Question

Please help I don’t understand

Answer 1

Answer: A) 35

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Short Explanation:

Subtract the arcs and divide by 2 like so

angle 1 = (longerArc - shorterArc)/2

angle 1 = (100 - 30)/2

angle 1 = 70/2

angle 1 = 35 degrees

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Longer explanation:

Refer to the diagram below. I have replaced the 100 and 30 degree arcs with A and B respectively. I've also added in the arc measures of y and z.

Note how the smaller circle arc pieces A, B, y and z all form a complete full circle. So those values add to 360. We get A+B+y+z = 360. We'll use this later.

In the diagram, I've also drawn a blue segment to form a triangle. The angle x we want to find has replaced angle 1. The angles (y+B)/2 and (z+B)/2 are found through the inscribed angle theorem.

Focus on the triangle. The three angles of any triangle always add to 180, so,

(z+B)/2 + (y+B)/2 + x = 180

we can multiply both sides by 2 and we end up with

z+B + y+B + 2x = 360

z+y+2B + 2x = 360

Since we have 360 on the right hand side of this equation, and the equation A+B+y+z = 360, this means that the left sides of each equation are the same.

We can then say the following

z+y+2B + 2x = A+B+y+z

2B+2x = A+B

2x = A+B-2B

2x = A-B

x = (A-B)/2

This is the formula used in the "short explanation" section above. In this case, A = 100 and B = 30, so

x = (A-B)/2

x = (100-30)/2

x = 70/2

x = 35

The measure of angle 1 is 35 degrees.

Side note: With that formula above, we always have A > B.

Answer 2

`m\angle 1 = 35^\circ`

Explanation:

Exterior Angle of a Circle

An exterior angle of a circle is an angle whose vertex is outside a circle and the sides of the angle are secants or tangents of the circle.

Please refer to the figure below.

The measure of the exterior angle x is given by:

`\displaystyle x=(b-a)/(2)`

Where a and b are the intercepted arcs.

We are given b=100° and a=30°, thus angle 1 is:

`\displaystyle m\angle a=(100°-30°)/(2)`

`\boxed{m\angle 1 = 35^\circ}`