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