Seeking Circle Angles with Answers
In this lesson it is your job to discover relationships that exist in circles.
Using the Exploration sheet “Circle Angle Examples” explore some relationships between arcs and angles
for angles that are inscribed within a circle, angles formed on the interior of circles, and angles formed
exterior to the circle.
Inscribed Angles
Using the first Circle examples, fill in the following chart and answer the questions.
Measure of Inscribed Angle
Measure of Intercepted Arc
74.5°
45°
149°
90°
82°
164°
15°
30°
What relationship do you notice between the measure of the inscribed angle and the measure of the
intercepted arc?
____Each Inscribed angle is half of the intercepted arc, or the intercepted arc is double the inscribed
angle._________________________________________________________________________
Can you generalize this relationship into a rule that will always work?
1
__𝑖𝑛𝑠𝑐𝑟𝑖𝑏𝑒𝑑 ∠ = 2 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑒𝑑 𝑎𝑟𝑐________________________________________
Angles formed in the Interior of a Circle
Using the second set of circles, investigate the angles formed in the interior of a circle.
Angle
Formed
Larger
Arc
Smaller
Arc
Sum of the
two Arcs
two Arcs)
Difference of
the two Arcs
88°
116°
60°
176°
88°
56°
28°
105°
118°
92°
210°
105°
26°
13°
120°
155°
85°
240°
120°
70°
35°
83°
126°
40°
166°
83°
86°
43°
1
2
(Sum of the
1
2
(Difference of
the two Arcs)
Do you notice a relationship between the arcs and the angle formed?
____Half of the sum of the two arcs is equal to the interior angle
______________________________________________________________________
Can you generalize this pattern into a rule that will always work?
Angle =
1
2
(𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑟𝑐𝑠)
Angles formed Exterior to the Circle
Angle
Formed
Larger
Arc
Smaller
Arc
Sum of the
two Arcs
1
(𝑆𝑢𝑚)
2
Difference of
the two Arcs
1
(𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒)
2
34°
110°
42°
152°
76°
68°
34°
54°
195°
87°
282°
141°
108°
54°
12°
37°
13°
50°
25°
24°
12°
50°
173°
73°
246°
123°
100°
50°
Do you notice a relationship between the arcs and the angle formed?
___The Angle is ½ of the difference of the two arcs.
_______________________________________________________________________
Can you generalize this pattern into a formula that will always work?
________ Angle =
1
2
(𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑟𝑐𝑠 )__________________________________
_______________________________________________________________________
Summary:
1
Inscribed angle = ___ (𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑒𝑑 𝑎𝑟𝑐) _________________________________
2
1
Angle formed in the interior = ____ (𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑟𝑐𝑠) _____________________
2
1
Angle formed in the exterior=_ (𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑟𝑐𝑠 _________________________
2