Respuesta :

choptsui

Answer:     <CAB = 39          arcEBC = 126

Step-by-step explanation:

Rule 1:   All the angles of a circle add up to equal 360

Rule 2:  The arc angle and the angle are the same if the angle starts at the center of the circle

Circle 1:

<CAD + <CAB + <BAE + <EAD = 360            >Add all angles and substitute

200 + <CAB + 73 + 48 = 360                         >Add numbers on left      

321 + <CAB = 360                                           >Subtract 321 from both sides  <CAB = 39

arcCDE = arcCAD + arcED

arc CDE = 200 + 48

arcCDE = 248

Circle 2:

Let's call the center O

Find: arcEBC

arcEBC = <EOC + <BOC

<DOE + <EOC + <BOC + <COD = 360         >Add all angles then substitute

123 + <EOC + 67 + 111 = 360                         >Add numbers on left  

<EOC + 301 = 360                                         >Subtract 301 from both sides

<EOC = 59

arcEBC = <EOC + <BOC

arcEBC = 59 + 67

arcEBC = 126

arcDCB = arcDC + arc CB

arcDCB = 111 + 67

arcDCB = 178

semsee45

Answer:

Arc CDE = 248°

∠CAB = 39°

Arc DCB = 178°

Arc EBC = 126°

Step-by-step explanation:

Question 1

An arc is named using its endpoints. If the arc is named using three letters, the middle letter is any other point contained in the arc.

Therefore, the arc CDE is the sum of arc CD and arc DE:

[tex]\begin{aligned}\overset{\frown}{CDE}&=\overset{\frown}{CD}+\overset{\frown}{DE}\\&=200^{\circ}+48^{\circ}\\&=248^{\circ}\end{aligned}[/tex]

Therefore, arc CDE measures 248°.

As the measure of an arc is equal to the measure of its corresponding central angle, and angles around a point sum to 360°, then:

[tex]\angle CAB+\angle CAD + \angle DAE + \angle EAD = 360^{\circ}[/tex]

[tex]\angle CAB+\overset{\frown}{CD} + \overset{\frown}{DE} + \angle EAD = 360^{\circ}[/tex]

[tex]\angle CAB+200^{\circ}+ 48^{\circ} + 73^{\circ} = 360^{\circ}[/tex]

[tex]\angle CAB+321^{\circ} = 360^{\circ}[/tex]

[tex]\angle CAB=39^{\circ}[/tex]

Therefore, angle CAB measures 39°.

[tex]\hrulefill[/tex]

Question 2

An arc is named using its endpoints. If the arc is named using three letters, the middle letter is any other point contained in the arc.

Therefore, the arc DCB is the sum of arc DC and arc CB:

[tex]\overset{\frown}{DCB}=\overset{\frown}{DC}+\overset{\frown}{CB}[/tex]

The measure of an arc is equal to the measure of its corresponding central angle. Therefore:

[tex]\begin{aligned}\overset{\frown}{DCB}&=\overset{\frown}{DC}+\overset{\frown}{CB}\\&=111^{\circ}+67^{\circ}\\&=178^{\circ}\end{aligned}[/tex]

Therefore, arc DCB measures 178°.

As the measure of an arc is equal to the measure of its corresponding central angle, and angles around a point sum to 360°, then:

[tex]\overset{\frown}{DC} + \overset{\frown}{CB} + \overset{\frown}{BE} +\overset{\frown}{ED}= 360^{\circ}[/tex]

[tex]111^{\circ} + 67^{\circ} + \overset{\frown}{BE} +123^{\circ}= 360^{\circ}[/tex]

[tex]\overset{\frown}{BE} +301^{\circ}= 360^{\circ}[/tex]

[tex]\overset{\frown}{BE} =59^{\circ}[/tex]

The arc EBC is the sum of arc EB and arc BC:

[tex]\begin{aligned}\overset{\frown}{EBC}&=\overset{\frown}{EB}+\overset{\frown}{BC}\\&=59^{\circ}+67^{\circ}\\&=126^{\circ}\end{aligned}[/tex]

Therefore, arc EBC measures 126°.