How to Study Central and Inscribed Angles of a Circle: Self Quiz 1 - Free Printable

I'll solve each of the four problems on this "Self Quiz" worksheet, which involves finding arc measures, angle measures, and solving for variables in circle geometry.

Let me start by examining each problem one by one.

Problem 1:
I need to find the measure of arc KL. I can see there's an inscribed angle JKL that measures 33°. According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc. So, arc KL should be twice the measure of angle JKL.

Problem 2:
I need to find the measure of angle G. I can see the measures of arcs EF (150°), FG (86°), and GH (110°). Since these are arcs of a circle, I can find the measure of arc HE by subtracting the sum of the other three arcs from 360°. Then, I can use the fact that angle G is an inscribed angle intercepting arc HE.

Problem 3:
I need to solve for x. I can see that angle ACB measures 59°, and arc AB is given as 11x + 8. Since angle ACB is an inscribed angle intercepting arc AB, I can set up an equation using the Inscribed Angle Theorem.

Problem 4:
I need to solve for x. I can see that angles W, Y, and Z are given as 82°, 84°, and 60° respectively, and angle X is given as -8 + 6x. Since WXYZ is a cyclic quadrilateral (inscribed in a circle), opposite angles are supplementary. So, angle W + angle Y = 180° and angle X + angle Z = 180°.

Let me now calculate the solutions for each problem.

Problem 1 Solution:
Angle JKL = 33°
Arc KL = 2 × 33° = 66°

Problem 2 Solution:
Sum of given arcs = 150° + 86° + 110° = 346°
Arc HE = 360° - 346° = 14°
Angle G = ½ × Arc HE = ½ × 14° = 7°

Problem 3 Solution:
Angle ACB = 59°
Arc AB = 11x + 8
Since angle ACB is an inscribed angle intercepting arc AB:
59° = ½ × (11x + 8)
Multiply both sides by 2:
118° = 11x + 8
Subtract 8 from both sides:
110° = 11x
Divide by 11:
x = 10

Problem 4 Solution:
In cyclic quadrilateral WXYZ, angle X + angle Z = 180°
Angle Z = 60°
Angle X = -8 + 6x
So: (-8 + 6x) + 60 = 180
Simplify: 52 + 6x = 180
Subtract 52 from both sides: 6x = 128
Divide by 6: x = 128/6 = 64/3 ≈ 21.33

Wait, let me double-check Problem 4. I should verify if the opposite angles are indeed supplementary. Let me check angle W + angle Y = 82° + 84° = 166°, which is not 180°. This suggests I may have misidentified the opposite angles or there's an error in my assumption.

Actually, looking at the diagram again, I think I made an error. In a cyclic quadrilateral, the sum of opposite angles is 180°. Let me identify the correct opposite angles. If the vertices are labeled W, X, Y, Z in order around the circle, then opposite angles would be W and Y, and X and Z.

Given: W = 82°, Y = 84°, Z = 60°, X = -8 + 6x
W + Y = 82° + 84° = 166° ≠ 180°
X + Z = (-8 + 6x) + 60 = 52 + 6x

This doesn't add up to 180° either unless 52 + 6x = 180, which gives x = 128/6 = 64/3.

But since W + Y = 166°, which is not 180°, there might be an error in the problem or my interpretation. However, since the problem asks to solve for x, and given that X + Z should equal 180° in a cyclic quadrilateral, I'll proceed with that assumption.

Therefore, the solutions are:
1) Arc KL = 66°
2) Angle G = 7°
3) x = 10
4) x = 64/3 or approximately 21.33

Let me provide the final answer.

The solutions to the problems are:
1) The measure of arc KL is 66°.
2) The measure of angle G is 7°.
3) The value of x is 10.
4) The value of x is 64/3 (or approximately 21.33).