warm-up
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Warm-up
1) Solve for x, y, and z.
2) Solve for x.
3) Solve for x.
Today’s Agenda
Review of Chapter 12 Theorems 12.4 Secants
angle measures segment proportions
Next Class Review/Test
Check Skyward
Missing Quizzes/Tests 1st
12 Quiz Sharlanae, Courtney, Lillian, Jordan,
Bridger, Johnny, Abigail 10 Test
Armin, Pouria, Jordan, 10 Quiz
Armin, Abigail
Missing Quizzes/Tests 5th
12 Quiz Josi, Conner P, Nikol
10 Test Josi, Andrew, Nikol
10 Test Shelby
Check Skyward
Missing Quizzes/Tests 6th
Chapter 12 Quiz Julian, Tanner C, Connor
Chapter 10 Test Sam, Connor, Hunter
Chapter 10 Quiz Coleman, Tanner R, Kolton
Check Skyward
Tangent Lines
A tangent to a circle is a line that intersects a circle at exactly one point.
The point of intersection is called the point of tangency.
Theorem 12-1
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
Theorem 12.2
Converse of 12.1 If a line is perpendicular to a radius at its
endpoint on the circle, then the line is tangent to the circle.
Theorem 12.3
2 segments tangent to a circle from a point outside the circle are congruent.
Theorem 12.4
Within a circle or in congruent circles… congruent central angles have congruent
chords
Angle DOB Angle COA DB CA
Theorem 12.4
Within a circle or in congruent circles… congruent chords have congruent arcs
DB CA Arc DB Arc CA
Theorem 12.4
Within a circle or in congruent circles… congruent arcs have congruent central
angles
Arc DB Arc CA Angle DOB Angle COA
Theorem 12.5
Within a circle or in congruent circles… chords equidistant from the center are
congruent (side note) measure distance with
perpendicular line
CL CM XW ZY
Theorem 12.5
Within a circle or in congruent circles… congruent chords are equidistant from
the center.
XW ZY CL CM
Theorem 12.6
In a circle, a diameter that is perpendicular to a chord bisects the chord and its arc.
Theorem 12.7
In a circle, a diameter that bisects a chord (that is not the diameter) is perpendicular to the chord.
Theorem 12.8
In a circle, the perpendicular bisector of a chord contains the center.
Inscribed Circle
Inscribed Angle Angle whose vertex is on a circle and
whose sides are chords. Intercepted arc
Arc created by an inscribed angle.
Theorem 12.9-Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
ABC = ½AC
Corollaries to the Inscribed Angle Theorem
1) Two inscribed angles that share an intercepted arc are congruent.
2) An angle inscribed by a semicircle is a right angle.
Corollaries to the Inscribed Angle Theorem
3) The opposite angles of a quadrilateral inscribed in a circle are supplementary.
angle N + angle O = 180˚ angle P + angle M = 180˚
Theorem 12.10
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Review
Identify the Following Chord Diameter Secant Line
Secant Lines
A secant line is a line that intersects 2 sides of a circle.
Is the diameter a secant?
Theorem 12.11 Part 1
The measure of an angle formed by 2 lines that intersect inside a circle is the average of the 2 arcs.
angle 1 =
Example 2
Find the value of x.
Theorem 12.11 Part 2
The measure of an angle formed by 2 lines that intersect outside a circle is the difference of the arcs divided by 2.
x is the bigger angle
Example 2
Find the value of x.
Theorem 12.12 Part 1
If two chords intersect, then .
Example 3a
Find the value of x.
Theorem 12.2 Part 2
If 2 secant segments intersect, then (w + x)w = (z + y)y
Example 3c
Find the value of x.
Theorem 12.2 part 3
If a secant segment and a tangent segment intersect, then (y + z)y = t2
Example 3b
Find the value of z.
Assignment
12-4 Worksheet Turn in CRT Review
Extra Credit pg 707 #1 – 21 all skip 5
Check off 12-3
Assignment Practice
Assignment Practice
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