Angles In A Circle Worksheet Awesome 14 Best Of Different Types ... - Free Printable
Educational worksheet: Angles In A Circle Worksheet Awesome 14 Best Of Different Types .... Download and print for classroom or home learning activities.
JPG
2550×3300
321.6 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #1723115
⭐
Show Answer Key & Explanations
Step-by-step solution for: Angles In A Circle Worksheet Awesome 14 Best Of Different Types ...
▼
Show Answer Key & Explanations
Step-by-step solution for: Angles In A Circle Worksheet Awesome 14 Best Of Different Types ...
Here is the step-by-step solution for each problem on the Geometry Review worksheet. The problems involve circle theorems, especially those relating to angles formed by chords, secants, tangents, and arcs.
---
Given: An angle formed outside a circle by two secants intersecting at point X. One intercepted arc is 100°, the other is 30°.
Find: x
> Theorem: The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs.
\[
x = \frac{1}{2} (100^\circ - 30^\circ) = \frac{1}{2}(70^\circ) = 35^\circ
\]
✔ Answer: x = 35°
---
Given: Two chords intersect inside a circle. One arc is 70°, another is 120°. Angle x is formed at the intersection.
Find: x
> Theorem: The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs.
\[
x = \frac{1}{2} (70^\circ + 120^\circ) = \frac{1}{2}(190^\circ) = 95^\circ
\]
✔ Answer: x = 95°
---
Given: A tangent and a secant intersect at point C outside the circle. The major arc AB is 240°.
Find: x (angle at C)
> Theorem: The measure of an angle formed by a tangent and a secant intersecting outside the circle is half the difference of the intercepted arcs.
First, find the minor arc AB:
\[
\text{Minor arc AB} = 360^\circ - 240^\circ = 120^\circ
\]
Now apply the theorem:
\[
x = \frac{1}{2} (240^\circ - 120^\circ) = \frac{1}{2}(120^\circ) = 60^\circ
\]
✔ Answer: x = 60°
---
Given: An angle formed outside the circle by a secant and a tangent. One intercepted arc is 130°, the other is 60°.
Find: x
> Theorem: Angle = ½ |difference of intercepted arcs|
\[
x = \frac{1}{2} (130^\circ - 60^\circ) = \frac{1}{2}(70^\circ) = 35^\circ
\]
✔ Answer: x = 35°
---
Given: A central angle and an inscribed angle sharing an arc. The arc is 110°, and there’s a vertical angle labeled x.
Find: x
> Observation: The diagram shows a central angle of 110° (since it's at the center). The angle labeled x appears to be an inscribed angle intercepting the same 110° arc.
> Theorem: Inscribed angle = ½ × intercepted arc
\[
x = \frac{1}{2} \times 110^\circ = 55^\circ
\]
✔ Answer: x = 55°
*(Note: If x were meant to be the central angle, it would be 110°, but based on position and typical notation, it’s likely the inscribed angle.)*
---
Given: Two chords intersect inside the circle. One arc is 50°, the angle at intersection is 40°. Find the unknown arc labeled x.
Find: x
> Theorem: Angle formed by intersecting chords = ½(sum of intercepted arcs)
Let the unknown arc be x.
\[
40^\circ = \frac{1}{2} (50^\circ + x)
\]
Multiply both sides by 2:
\[
80^\circ = 50^\circ + x
\Rightarrow x = 30^\circ
\]
✔ Answer: x = 30°
---
Given: An inscribed angle x intercepting an arc that is part of a circle. The two arcs shown are 140° and 100°. The angle x is formed by chords from endpoints of these arcs.
Find: x
> Observation: The angle x is an inscribed angle. The intercepted arc is the one *not* including the 140° and 100°? Wait — actually, in this diagram, the angle x is formed by two chords that intercept the arc opposite to it.
Actually, the total circle is 360°. The two given arcs are 140° and 100°, so the third arc (the one intercepted by angle x) is:
\[
360^\circ - 140^\circ - 100^\circ = 120^\circ
\]
> Theorem: Inscribed angle = ½ × intercepted arc
\[
x = \frac{1}{2} \times 120^\circ = 60^\circ
\]
✔ Answer: x = 60°
---
Given: A tangent and a diameter intersect at a point on the circle. The arc between them is labeled \(2x + 10^\circ\).
Find: x
> Key Insight: The tangent is perpendicular to the radius at the point of contact. Since the diameter passes through the center, the angle between the tangent and the diameter is 90°.
But here, we’re given an arc measure: \(2x + 10^\circ\). This arc is the one “between” the tangent and the diameter — which must be the arc intercepted by the right angle formed.
Wait — actually, the arc subtended by a 90° angle at the circumference would be 180°, but that’s not matching.
Let’s re-think:
The diagram shows a horizontal diameter and a vertical tangent line meeting at a point on the circle. The angle between them is 90°, and the arc between the two points where the lines meet the circle is labeled \(2x + 10^\circ\).
Actually, since the tangent and diameter meet at 90°, and the arc they cut off is a semicircle? No — wait.
Actually, the arc labeled \(2x + 10^\circ\) is likely the arc that is intercepted by the angle formed by the tangent and the diameter — which is 90°.
But according to circle theorems, the measure of an inscribed angle is half the intercepted arc. However, here the angle is formed by a tangent and a chord (diameter), which is a special case.
> Theorem: The angle between a tangent and a chord is equal to half the measure of the intercepted arc.
In this case, the angle between the tangent and the diameter is 90°, so:
\[
90^\circ = \frac{1}{2} \times \text{intercepted arc}
\Rightarrow \text{intercepted arc} = 180^\circ
\]
But the problem labels the arc as \(2x + 10^\circ\), so:
\[
2x + 10 = 180
\Rightarrow 2x = 170
\Rightarrow x = 85
\]
✔ Answer: x = 85
---
## ✔ Final Answers:
1. x = 35°
2. x = 95°
3. x = 60°
4. x = 35°
5. x = 55°
6. x = 30°
7. x = 60°
8. x = 85
---
Let me know if you’d like diagrams or further explanation for any problem!
---
Problem 1
Given: An angle formed outside a circle by two secants intersecting at point X. One intercepted arc is 100°, the other is 30°.
Find: x
> Theorem: The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs.
\[
x = \frac{1}{2} (100^\circ - 30^\circ) = \frac{1}{2}(70^\circ) = 35^\circ
\]
✔ Answer: x = 35°
---
Problem 2
Given: Two chords intersect inside a circle. One arc is 70°, another is 120°. Angle x is formed at the intersection.
Find: x
> Theorem: The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs.
\[
x = \frac{1}{2} (70^\circ + 120^\circ) = \frac{1}{2}(190^\circ) = 95^\circ
\]
✔ Answer: x = 95°
---
Problem 3
Given: A tangent and a secant intersect at point C outside the circle. The major arc AB is 240°.
Find: x (angle at C)
> Theorem: The measure of an angle formed by a tangent and a secant intersecting outside the circle is half the difference of the intercepted arcs.
First, find the minor arc AB:
\[
\text{Minor arc AB} = 360^\circ - 240^\circ = 120^\circ
\]
Now apply the theorem:
\[
x = \frac{1}{2} (240^\circ - 120^\circ) = \frac{1}{2}(120^\circ) = 60^\circ
\]
✔ Answer: x = 60°
---
Problem 4
Given: An angle formed outside the circle by a secant and a tangent. One intercepted arc is 130°, the other is 60°.
Find: x
> Theorem: Angle = ½ |difference of intercepted arcs|
\[
x = \frac{1}{2} (130^\circ - 60^\circ) = \frac{1}{2}(70^\circ) = 35^\circ
\]
✔ Answer: x = 35°
---
Problem 5
Given: A central angle and an inscribed angle sharing an arc. The arc is 110°, and there’s a vertical angle labeled x.
Find: x
> Observation: The diagram shows a central angle of 110° (since it's at the center). The angle labeled x appears to be an inscribed angle intercepting the same 110° arc.
> Theorem: Inscribed angle = ½ × intercepted arc
\[
x = \frac{1}{2} \times 110^\circ = 55^\circ
\]
✔ Answer: x = 55°
*(Note: If x were meant to be the central angle, it would be 110°, but based on position and typical notation, it’s likely the inscribed angle.)*
---
Problem 6
Given: Two chords intersect inside the circle. One arc is 50°, the angle at intersection is 40°. Find the unknown arc labeled x.
Find: x
> Theorem: Angle formed by intersecting chords = ½(sum of intercepted arcs)
Let the unknown arc be x.
\[
40^\circ = \frac{1}{2} (50^\circ + x)
\]
Multiply both sides by 2:
\[
80^\circ = 50^\circ + x
\Rightarrow x = 30^\circ
\]
✔ Answer: x = 30°
---
Problem 7
Given: An inscribed angle x intercepting an arc that is part of a circle. The two arcs shown are 140° and 100°. The angle x is formed by chords from endpoints of these arcs.
Find: x
> Observation: The angle x is an inscribed angle. The intercepted arc is the one *not* including the 140° and 100°? Wait — actually, in this diagram, the angle x is formed by two chords that intercept the arc opposite to it.
Actually, the total circle is 360°. The two given arcs are 140° and 100°, so the third arc (the one intercepted by angle x) is:
\[
360^\circ - 140^\circ - 100^\circ = 120^\circ
\]
> Theorem: Inscribed angle = ½ × intercepted arc
\[
x = \frac{1}{2} \times 120^\circ = 60^\circ
\]
✔ Answer: x = 60°
---
Problem 8
Given: A tangent and a diameter intersect at a point on the circle. The arc between them is labeled \(2x + 10^\circ\).
Find: x
> Key Insight: The tangent is perpendicular to the radius at the point of contact. Since the diameter passes through the center, the angle between the tangent and the diameter is 90°.
But here, we’re given an arc measure: \(2x + 10^\circ\). This arc is the one “between” the tangent and the diameter — which must be the arc intercepted by the right angle formed.
Wait — actually, the arc subtended by a 90° angle at the circumference would be 180°, but that’s not matching.
Let’s re-think:
The diagram shows a horizontal diameter and a vertical tangent line meeting at a point on the circle. The angle between them is 90°, and the arc between the two points where the lines meet the circle is labeled \(2x + 10^\circ\).
Actually, since the tangent and diameter meet at 90°, and the arc they cut off is a semicircle? No — wait.
Actually, the arc labeled \(2x + 10^\circ\) is likely the arc that is intercepted by the angle formed by the tangent and the diameter — which is 90°.
But according to circle theorems, the measure of an inscribed angle is half the intercepted arc. However, here the angle is formed by a tangent and a chord (diameter), which is a special case.
> Theorem: The angle between a tangent and a chord is equal to half the measure of the intercepted arc.
In this case, the angle between the tangent and the diameter is 90°, so:
\[
90^\circ = \frac{1}{2} \times \text{intercepted arc}
\Rightarrow \text{intercepted arc} = 180^\circ
\]
But the problem labels the arc as \(2x + 10^\circ\), so:
\[
2x + 10 = 180
\Rightarrow 2x = 170
\Rightarrow x = 85
\]
✔ Answer: x = 85
---
## ✔ Final Answers:
1. x = 35°
2. x = 95°
3. x = 60°
4. x = 35°
5. x = 55°
6. x = 30°
7. x = 60°
8. x = 85
---
Let me know if you’d like diagrams or further explanation for any problem!
Parent Tip: Review the logic above to help your child master the concept of geometry circles worksheet.
