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Use Inscribed Angles and Polygons. Wednesday, September 17, 2014. How do we use inscribed angles of circles?. Lesson 6.4. M2 Unit 3: Day 4. Find the value of x in C . Explain. 1. ANSWER. - PowerPoint PPT Presentation
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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.
Use Inscribed Angles and Polygons
How do we use inscribed angles of circles?
M2 Unit 3: Day 4
Lesson 6.4
Friday, April 21, 2023
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
1.
Find the value of x in C. Explain. .
ANSWER
6; If a diameter of a circle is to the chord, then the diameter bisects the chord and its arc.
Daily Homework Quiz
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
2.
Find the value of x in C. Explain. .
ANSWER
4; In the same circle, if two chords are equidistant from the center, then they are .=~
Daily Homework Quiz
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
3. Determine whether RS is a diameter.
ANSWER
Yes. Sample answer: RS is the bisector of TU by Theorem 5.3. Then RS is a diameter of the circle by Theorem 10.4.
Daily Homework Quiz
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.
ANSWER 25
1. The measure of the interior angles of a quadrilateralare 80º, 100º, 55º, and 5xº. Find the value of x.
2. Two supplementary angles have measures 6xº and12xº. Find each angle measure.
ANSWER 60º; 120º
Warm Ups
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. Solve 3x = ( 4x + 12).1 2
4. Solve 80 = ( 360 – 2x).1 2
Warm Ups
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
Inscribed AngleInscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).
1 42 3
No! No!Yes! Yes!
Examples:
Inscribed Angle
Intercepted Arc
B
A
CCB is an inscribed angleA
» is the intercepted arc of ÐAB C
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.
Theorem 6.9 Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
»1BC
2Ð =m A mAC B
C
A
O
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 Use inscribed angles
a. m T mQRb.
Find the indicated measure in P.
SOLUTION
12
12
M T = mRS = (48o) = 24oa.
mQR = 180o mTQ = 180o 100o = 80o. So, mQR = 80o. – –
mTQ = 2m R = 2 50o = 100o. Because TQR is a semicircle,b.
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 measure of the red arc or angle.
1.
SOLUTION
12
12
M G = mHF = (90o) = 45oa.
Guided Practice
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 measure of the red arc or angle.
2.
SOLUTION
mTV = 2m U = 2 38o = 76o. b.
Guided Practice
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.
Theorem 6.10
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
A
B
D
C
ABD ACD
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 measure of an intercepted arc
Find mRS and m STR. What do you notice about STR and RUS?
SOLUTION
From Theorem 6.9, you know that mRS = 2m RUS = 2 (31o) = 62o.
Also, m STR = mRS = (62o) = 31o. So, STR RUS.12
12
EXAMPLE
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 3
Standardized Test Practice
SOLUTION
Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 6.10 Also, KJL and KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.
So, by Theorem 10.8, the correct answer is C.
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 measure of the red arc or angle.
3.
SOLUTION
ZYN ZXN
ZXN 72°
Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem 6.10. Also, KJL and KML intercept the same arc, so they must also be congruent.
ZXN
Guided Practice
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 mLJM.
mLJM = 5(3.5) – 7 = 10.5 Substitute 3.5 for b.
5b – 7 = 3b Substitute the given values.
2b – 7 = 0 Subtract 3b from both sides.
2b = 7 Add 7 to both sides.
b = 3.5 Divide both sides by 2.
mLJM = mLKM mLJM and mLKM
both intercept LM.
Guided Practice
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 mEDF.
2x + 3 = 75 – 2x Substitute the given values.
4x = 72 Add 2x and subtract 3 from both sides.
x = 18 Divide both sides by 4.
mEDF = 2(18) + 3 = 39°
mEDF = mEGF mEGF and mEDF
both intercept EF.
Guided Practice
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.
18
An angle inscribed in a semicircle is a right angle.
R
P 180
S90
Theorem 6.11 Inscribed Right Angle Theorem
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.
Find a.
WZY is a right angle WZY is inscribed in a semicircle.
mWZY = 90 Def of rt.
5a + 20 = 90 Substitute 5a + 20 for mWZY.
5a = 70 Subtract 20 from both sides.
a = 14 Divide both sides by 5.
EXAMPLE
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.
6. Find z.
8z – 6 = 90 Substitute.
8z = 96 Add 6 to both sides.
z = 12 Divide both sides by 8.
ABC is a right angle ABC is inscribed in a semicircle.
mABC = 90 Def of rt.
Guided Practice
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.
21
mDAB + mDCB = 180
mADC + mABC = 180
If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.
AB
C
D
Theorem 6.12 Inscribed Quadrilateral Theorem
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 5
Find the value of each variable.
a.
SOLUTION
PQRS is inscribed in a circle, so opposite angles are supplementary.
a.
m P + m R = 180o
75o + yo = 180o
y = 105
m Q + m S = 180o
80o + xo = 180o
x = 100
EXAMPLE
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 5
b. JKLM is inscribed in a circle, so opposite angles are supplementary.
m J + m L = 180o
2ao + 2ao = 180o
a = 45
m K + m M = 180o
4bo + 2bo = 180o
b = 30
4a = 180 6b = 180
Find the value of each variable.
b.
SOLUTION
EXAMPLE
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
7.
Find the value of each variable.
SOLUTION
ABCD is inscribed in a circle, so opposite angles are supplementary.
m A + m C = 180o
y°+ 68o = 180o
y = 112
m D+ m B = 180o
82o + xo = 180o
x = 98
Guided Practice
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
8.
Find the value of each variable.
SOLUTION
STUV is inscribed in a circle, so opposite angles are supplementary.
m S + m U = 180o
c°+ (2c – 6 )o = 180o
c = 62
m V+ m T = 180o
8xo + 10xo = 180o
x = 10
(3c)o = 186o 18xo = 180o
Guided Practice
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.
9. Find the angle measures of GHJK.
mG + mJ = 180 GHJK is inscribed in a .
3b + 25 + 6b + 20 = 180 Substitute the given values.
9b + 45 = 180 Simplify.
9b = 135 Subtract 45 from both sides.
b = 15 Divide both sides by 9.
mG = 3(15) + 25 = 70 Substitute 15 for b
mJ = 6(15) + 20 = 110 in each expression.
mK = 10(15) – 69 = 81mH + mK = 180 H and K are supp.
mH + 81 = 180 Substitute 81 for mK.
mH = 99 Subtract 81 from both sides
Guided Practice
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.
Step 2 Find the measure of each angle.
Example 4 Continued
mG = 3(15) + 25 = 70 Substitute 15 for b
mJ = 6(15) + 20 = 110 in each expression.
mK = 10(15) – 69 = 81mH + mK = 180 H and K are supp.
mH + 81 = 180 Substitute 81 for mK.
mH = 99 Subtract 81 from both sides
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.
35o
xo
yoQ
L
K
J Find x and y.
90 QJL m 90o
55x
125180 y
1 2
m y JQ
35y
Guided Practice
70o
110o
180o
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: pg 207-208 (#1, 2, 5, 6, 8-18, 22)