Apply Other Angle Relationships in Circles
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Transcript of Apply Other Angle Relationships in Circles
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Apply Other Angle Relationships in Circles
Essential Question:
How do we find the measures of angles inside or outside a circle?
M2 Unit 3: Day 5Lesson 6.5
Friday, April 21, 2023
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.Daily Homework Quiz
Find the value of x .
1.
Daily Homework Quiz
2.
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
3.
Find the value of x .
Daily Homework Quiz
4.
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
5.
Find the value of each variable.
ANSWER x = 5; y = 10
Daily Homework Quiz
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
ANSWER 58
2. x = ( 360 – 120) = 01 2
1. 2x = 84 + 32
Solve for x.
Warm Ups
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
3. 180 – x = (( 2x + 4) + 28).1 2
4. One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle.
Solve for x.
ANSWER 45º
Warm Ups
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.13If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
21
C
A
B
m 2 = 1
2 m BCA
m 1 = 1
2 m AB
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
EXAMPLE Find angle and arc measures
Line m is tangent to the circle. Find the measure of the red angle or arc.
SOLUTION
= 12 (130o) = 65o
= 2 (125o) = 250ob. m KJL
a. m 1
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.GUIDED PRACTICE
Find the indicated measure.
SOLUTION
= 12 (210o) = 105om 1
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.GUIDED PRACTICE
Find the indicated measure.
= 2 (98o) = 196om TSR
SOLUTION
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.GUIDED PRACTICE
Find the indicated measure.
= 2 (80o) = 160om XY
SOLUTION
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.14 Angle Inside the CircleIf two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures
of the arc intercepted by the angle and its vertical angle.
1
A
C
B
D
m 1 = 1
2( mCD + mAB )
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.EXAMPLE 2
Find the value of x.
SOLUTION
The chords JL and KM intersect inside the circle.
Use Theorem 6.14.xo = 12
(mJM + mLK)
xo = 12
(130o + 156o) Substitute.
xo = 143 Simplify.
EXAMPLE
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
4. Find the value of the variable.
SOLUTION
The chords AC and CD intersect inside the circle.
Use Theorem 6.14
Substitute.
Simplify.
= 12
(yo + 95o) 78o
= y 61
78° (mAB + mCD)= 12
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.15Angle Outside the Circle
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.15 Secant and Tangent
1
A
B
C
m 1 = 1
2( m BC - m AC )
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.15Two Tangents
2
R
P
Q
m 2 = 1
2( m PQR - m PR )
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
Theorem 6.15 Two Secants
3
Z
W
X
Y
m 3 = 1
2( m XY - m WZ )
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
EXAMPLE Find an angle measure outside a circle
Find the value of x.
SOLUTION
Use Theorem 10.13.
Substitute.
Simplify.
The tangent CD and the secant CB intersect outside the circle.
= 12
(178o – 76o)xo
= 51 x
m BCD (mAD – mBD)= 12
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
= 40 = 63
Find the value of x.
EXAMPLE Find an angle measure outside a circle
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.GUIDED PRACTICE
Find the value of the variable.5.
SOLUTION
The tangent JF and the secant JG intersect outside the circle.
Use Theorem 10.13.
Substitute.
Simplify.
= 12
(ao – 44o) 30o
= 104 a
m FJG (mFG – mKH)= 12
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.GUIDED PRACTICE
6. Find the value of the variable.
SOLUTION
Use Theorem 10.13.
Substitute.73.7o 12
[(xo) –(360 –x)o]
Solve for x.xo 253.7
= 12
m TQR (mTUR – mTR)
Because QT and QR are tangents, QR RS and QT TS
Also,TS SR and CA CA . So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and TQS RQS.Solve right QTS to find that m RQS 73.7°.
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
7. Find the value of x.
50° = 83° – x
x = 33°
Guided Practice
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.EXAMPLE 4
Solve a real-world problemSCIENCE
The Northern Lights are bright flashes of colored light between 50 and 200 miles above Earth. Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion of Earth from which the flash is visible? (Earth’s radius is approximately 4000 miles.)
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.EXAMPLE 4
Solve a real-world problemSOLUTION
Use Theorem 10.13.
Substitute.149o 12
[(360o – xo) –xo]
Solve for x.xo 31
= 12
m BCD (mDEB – mBD)
Because CB and CD are tangents, CB AB and CD AD
Also,BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA.Solve right CBA to find that m BCA 74.5°.
ANSWERThe measure of the arc from which the flash is visible is about 31o.
MM2G3 Students will understand properties of circles.
MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties.
MM2G3 Students will understand properties of circles.
Homework
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