Download Inscribed Angle Inscribed Angle Theorems

Inscribed Angle
First off, a definition:
Inscribed Angle: an angle made from points sitting on the circle's circumference.
A and C are "end points"
B is the "apex point"
Inscribed Angle Theorems
An inscibed angle a° is half of the central angle 2a°
(Called the Angle at the Center Theorem)
And (keeping the endpoints fixed) ...
... the angle a° is always the same, no matter where it is on the circumference:
Angle a° is the same.
(Called the Angles Subtended by Same Arc Theorem)
Example: What is the size of Angle POQ? (O is circle's center)
Angle POQ = 2 × Angle PRQ = 2 × 62° = 124°
Example: What is the size of Angle CBX?
Angle ADB = 32° equals Angle ACB.
And Angle ACB equals Angle XCB.
So in triangle BXC we know Angle BXC = 85°, and Angle XCB =
32°
Now use angles of a triangle add to 180° :
Angle CBX + Angle BXC + Angle XCB = 180°
Angle CBX + 85° + 32° = 180°
Angle CBX = 63°
Angle in a Semicircle
An angle inscribed in a semicircle is always a right angle:
(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)
Why? Because:
The inscibed angle 90° is half of the central angle 180°
(Using "Angle at the Center Theorem" above)
Another Good Reason Why It
Works
We could also rotate the shape
around 180° to make a rectangle!
It is a rectangle, because all sides are
parallel, and both diagonals are
equal.
And so its internal angles are all
right angles (90°).
So there you go! No matter where that angle is
on the circumference, it is always 90°
Example: What is the size of Angle BAC?
The Angle in the Semicircle Theorem tells us that Angle ACB =
90°
Now use angles of a triangle add to 180° to find Angle BAC:
Angle BAC + 55° + 90° = 180°
Angle BAC = 35°
Cyclic Quadrilateral
A "Cyclic" Quadrilateral has every vertex on a circle's
circumference:
A Cyclic Quadrilateral's opposite angles add to 180°:


a + c = 180°
b + d = 180°
Example: What is the size of Angle WXY?
Opposite angles of a cyclic quadrilateral add to 180°
Angle WZY + Angle WXY = 180°
69° + Angle WXY = 180°
Angle WXY = 111°
Tangent Theorems
Tangent Angle
A tangent is a line that just touches a circle at one point.
It always forms a right angle with the circle's radius as
shown here.
Tangent Segments
Given a point outside a circle, two lines can be drawn through that point that are tangent to the
circle. The tangent segments whose endpoints are the points of tangency and the fixed point
outside the circle are equal. In other words, tangent segments drawn to the same circle from the
same point (there are two for every circle) are equal.
Figure %: Tangent segments that share an endpoint not on the circle are equal
Chords
Chords within a circle can be related many ways. Parallel chords in the same circle always cut
congruent arcs. That is, the arcs whose endpoints include one endpoint from each chord have
equal measures.
Figure %: Arcs AC and BD have equal measures
When congruent chords are in the same circle, they are equidistant from the center.
Figure %: Congruent chords in the same circle are equidistant from the center
In the figure above, chords WX and YZ are congruent. Therefore, their distances from the center, the
lengths of segments LC and MC, are equal.
A final word on chords: Chords of the same length in the same circle cut congruent arcs. That is,
if the endpoints of one chord are the endpoints of one arc, then the two arcs defined by the two
congruent chords in the same circle are congruent.
Intersecting Chords, Tangents, and Secants
A number of interesting theorems arise from the relationships between chords, secant segments,
and tangent segments that intersect. First of all, we must define a secant segment. A secant
segment is a segment with one endpoint on a circle, one endpoint outside the circle, and one
point between these points that intersects the circle. Three theorems exist concerning the above
segments.
Theorem 1
PARGRAPH When two chords of the same circle intersect, each chord is divided into two
segments by the other chord. The product of the segments of one chord is equal to the product of
the segments of the other chord.
Figure %: Chords of the same circle that intersect
In the figure above, chords QR and ST intersect. The theorem states that the product of QB and
BR is equal to the product of SB and BT.
Theorem 2
Every secant segment is divided into two segments by the circle it intersects. The internal
segment is a chord, and the external segment is the segment with one endpoint at the intersection
of the secant segment and the circle, and the other endpoint at the fixed point outside the circle.
Given these conditions, a theorem states that when two secant segments share an endpoint not on
the circle, the products of the lengths of each secant segment and its external segment are equal.
Figure %: Secant segments that share an endpoint not on the circle
In the figure above, the secant segments DE and FE share an endpoint, E, outside the circle. The
theorem states that the product of the lengths of DE and ME is equal to the product of the lengths of FE
and NE.
Theorem 3
A similar theorem exists when a secant segment and a tangent segment share an endpoint not on
the circle. This theorem states that the length of the tangent segment squared is equal to the
product of the secant segment and its external segment.
Figure %: A secant segment and a tangent segment that share an endpoint not on the circle
In the figure above, secant segment QR and tangent segment SR share an endpoint, R, not on the circle.
The theorem states that the length of SR squared is equal to the product of the lengths of QR and KR.