warm-up exercises answer 25 1. the measure of the interior angles of a quadrilateral are 80º,...

Warm-Up Exercises

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-Up Exercises

ANSWER 6

ANSWER 100

3. Solve 3x = ( 4x + 12).1 2

4. Solve 80 = ( 360 – 2x).1 2

Warm-Up ExercisesEXAMPLE 1 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.

Warm-Up ExercisesEXAMPLE 2 Find the measure of an intercepted arc

Find mRS and m STR. What do you notice about STR and RUS?

SOLUTION

From Theorem 10.7, you know that mRS = 2m RUS = 2 (31o) = 62o.

Also, m STR = mRS = (62o) = 31o. So, STR RUS.12

12

Warm-Up ExercisesEXAMPLE 3 Standardized Test Practice

SOLUTION

Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 10.8. 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.

Warm-Up ExercisesGUIDED PRACTICE for Examples 1, 2 and 3

Find the measure of the red arc or angle.

1.

SOLUTION

m G = mHF = (90o) = 45o12

12

a.



2.

SOLUTION

mTV = 2m U = 2 38o = 76o. b.



3.

SOLUTION

ZYN ZXN

ZXN 72°

Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem 10.8. Also, KJL and KML intercept the same arc, so they must also be congruent.

ZXN

Warm-Up ExercisesEXAMPLE 4Use a circumscribed circle

PhotographyYour camera has a 90o field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way?

Warm-Up ExercisesEXAMPLE 4Use a circumscribed circle

SOLUTION

From Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera’s 90o field of vision from any point on the semicircle in front of the statue.

Warm-Up ExercisesGUIDED PRACTICE for Example 4

What If ? In Example 4, explain how to find locations if you want to frame the front and left side of the statue in your picture.

4.

Make the diameter of your circle the diagonal of the rectangular base.

SOLUTION

Warm-Up ExercisesEXAMPLE 5 Use Theorem 10.10

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

Warm-Up ExercisesEXAMPLE 5 Use Theorem 10.10

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


5.

Find the value of each variable.

SOLUTION

y = 112 x = 98


6.


SOLUTION

c = 62 x = 10

Warm-Up ExercisesDaily Homework Quiz

ANSWER 38

Find the value of x .

1.


2.


ANSWER 56


3.


ANSWER 44


4.


ANSWER x = 54; y = 20


5.


ANSWER x = 5; y = 10

warm-up exercises answer 25 1. the measure of the interior angles of a quadrilateral are 80º,...

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