Download 14.3 Secant Angles

Geometry – Chapter 14 Lesson Plans
Section 14.3 – Secant Angles
Enduring Understandings: The student shall be able to:
1. Find measure of arcs and angles formed by secants.
Standards:
30. Circles
Identifies and defines circles and their parts (center, arc, interior, exterior); segments
and lines associated with circles (chord, diameter, radius, tangent, secant); properties
of circles (congruent, concentric, tangent); relationship of polygons and circles
(inscribed, circumscribed); angles (central; inscribed; formed by tangents, chords, and
secants).
Essential Questions: What is a secant angle, and how do we measure it?
Warm up/Opener:
Activities:
A secant is a line that intersects the circle in two points. Compare and contrast a secant
and chord. A chord is a secant segment.
A secant angle is the angle between two intersecting secants.
There are three cases of secant angles:
1. The vertex is on the circle.
The measure of the vertex angle is one-half the measure of the intercepted arc.
2. The vertex is inside the circle.
Thm 14.8: If a secant angle has its vertex inside a circle, then its degree measure
is one-half the sum of the degree measures of the arcs intercepted by the angle
and its vertical angle.
3. The vertex is outside the circle.
Thm 14.9: If a secant angle has its vertex outside a circle, then its degree measure
is one-half the difference of the degree measures of the intercepted arcs.
 Proof for #2 above, vertex Inside the circle
Construct segment BC.
Angle 1 is an exterior angle of  BCP, so m  1 = m  B + m  C
But m  B = ½ m arc DC and m  C = ½ m arc AB, so
m1 = ½ m arc DC + ½ m arc AB = ½ (m arc DC + m arc AB)
D
P
A
1
B
C
 Proof for #3 above, Vertex is Outside the circle
Construct segment BC.
1 = ½ m arc BD =  A + B = A + ½ m arc EC
A = ½ (m arc BD – m arc EC)
B
E
A
D
C
1
Do the “In-Class” examples in the blue book, and the three “Your Turn” problems.
Assessments:
Do the “Check for Understanding” 1, 2, 4-8
CW WS 14.3
HW pg 604-605, # 9-31 all (22 – if I only do this section as a class)
HW pg 604-605, # 9 - 31 odd (11 – if I combine this with another section)
Extra Credit: Enrichment 14-3