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Circle Geometry Theorems
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Circle Geometry Theorems
Circle Geometry Theorems
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1
Question
What is a chord of a circle?
Answer
A straight line that joins two points on the circumference of the circle.
2
Question
How does a chord divide a circle?
Answer
A chord splits the circle into two segments.
3
Question
What is a diameter in the context of a circle?
Answer
A chord that passes straight through the center of the circle.
4
Question
What defines a tangent to a circle?
Answer
A straight line that touches the circle at exactly one point.
5
Question
What is a cyclic quadrilateral?
Answer
A quadrilateral where all four vertices lie on the circumference of a circle.
6
Question
What condition must fail for a quadrilateral not to be cyclic?
Answer
If one or more corners are not on the circumference of the circle.
7
Question
What does the angles in the same segment theorem state?
Answer
Angles subtended by the same chord in the same segment of the circle are equal.
8
Question
If one angle in the same segment measures 68 degrees, what measure do the other angles in that segment have?
Answer
68 degrees each.
9
Question
What diagram shape is sometimes associated informally with the angles in the same segment theorem?
Answer
A bow tie shape formed by the two angles and the arc.
10
Question
What is the angle in a semicircle theorem?
Answer
The angle subtended by a diameter at the circumference is 90 degrees.
11
Question
Why are angles subtended by a diameter in the same segment always 90 degrees?
Answer
Because the diameter splits the circle into a semicircle, making those angles right angles.
12
Question
What is the relationship between the angle at the center and the angle at the circumference for the same chord?
Answer
The angle at the center is twice the angle at the circumference.
13
Question
If the angle at the circumference subtended by a chord is 50 degrees, what is the angle at the center?
Answer
100 degrees.
14
Question
What property do opposite angles have in a cyclic quadrilateral?
Answer
They sum to 180 degrees.
15
Question
If one opposite angle in a cyclic quadrilateral is 86 degrees, what is the other opposite angle?
Answer
94 degrees.
16
Question
If one opposite angle in a cyclic quadrilateral is 110 degrees, what is the other?
Answer
70 degrees.
17
Question
What is the angle between a tangent and the radius to the point of contact?
Answer
90 degrees.
18
Question
What property do two tangents drawn from the same external point to a circle share?
Answer
They are equal in length.
19
Question
What does the line from the circle's center to the external point do to the angle between two tangents?
Answer
It bisects the angle.
20
Question
If the angle between two tangents from an external point is 30 degrees, what is each half after bisection by the center line?
Answer
15 degrees.
21
Question
What is the alternate segment theorem?
Answer
The angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment.
22
Question
How does the alternate segment theorem apply to angles on both sides of a chord with a tangent?
Answer
Each tangent-chord angle equals the angle in the alternate segment for that chord.
23
Question
In a diagram where angles ACD and ACE form a straight line with ACE at 105 degrees, what is angle ACD?
Answer
75 degrees.
24
Question
How is angle ABD related to angle ACD in the first example problem using circle theorems?
Answer
They are equal because they are angles in the same segment.
25
Question
In an isosceles triangle ACD with AD equal to DC and base angle ACD of 35 degrees, what is the other base angle?
Answer
35 degrees.
26
Question
What is angle ADC in isosceles triangle ACD with base angles of 35 degrees each?
Answer
110 degrees.
27
Question
In cyclic quadrilateral ABCD with angle ADC at 110 degrees, what is opposite angle ABC?
Answer
70 degrees.
28
Question
Why is angle BCA 90 degrees when AB is the diameter in the second problem?
Answer
Because it is the angle in a semicircle.
29
Question
In triangle ABC with angles ABC at 70 degrees and BCA at 90 degrees, what is angle CAB?
Answer
20 degrees.
30
Question
How is angle BCF related to angle CAB in the second problem?
Answer
They are equal by the alternate segment theorem.