Question

example 4
$overline{pr}$ and $overline{qt}$ are diameters of $odot a$. find each measure.

1. $mwidehat{uq}$
2. $mwidehat{pqr}$
3. $mwidehat{us}$
4. $mwidehat{rs}$
5. $mwidehat{ru}$
6. $mwidehat{sp}$
7. $mwidehat{pqs}$
8. $mwidehat{pru}$

Explanation:

Step1: Confirm full circle is 360°

A circle's total measure is $360^\circ$.

Step2: Find missing arc $\overset{\frown}{UT}$

Sum given arcs: $40^\circ+90^\circ+90^\circ+50^\circ+40^\circ=310^\circ$.
$m\overset{\frown}{UT}=360^\circ-310^\circ=50^\circ$

Step3: Calculate $m\overset{\frown}{UQ}$ (12)

Add $\overset{\frown}{UT}+\overset{\frown}{TS}+\overset{\frown}{SR}+\overset{\frown}{RQ}$
$m\overset{\frown}{UQ}=50^\circ+40^\circ+50^\circ+90^\circ=230^\circ$

Step4: Calculate $m\overset{\frown}{PQR}$ (13)

Add $\overset{\frown}{PQ}+\overset{\frown}{QR}$
$m\overset{\frown}{PQR}=90^\circ+90^\circ=180^\circ$

Step5: Calculate $m\overset{\frown}{US}$ (14)

Add $\overset{\frown}{UT}+\overset{\frown}{TS}$
$m\overset{\frown}{US}=50^\circ+40^\circ=90^\circ$

Step6: Calculate $m\overset{\frown}{RS}$ (15)

Given directly on diagram
$m\overset{\frown}{RS}=50^\circ$

Step7: Calculate $m\overset{\frown}{RU}$ (16)

Add $\overset{\frown}{RS}+\overset{\frown}{ST}+\overset{\frown}{TU}$
$m\overset{\frown}{RU}=50^\circ+40^\circ+50^\circ=140^\circ$

Step8: Calculate $m\overset{\frown}{SP}$ (17)

Add $\overset{\frown}{ST}+\overset{\frown}{TU}+\overset{\frown}{UP}$
$m\overset{\frown}{SP}=40^\circ+50^\circ+40^\circ=130^\circ$

Step9: Calculate $m\overset{\frown}{PQS}$ (18)

Add $\overset{\frown}{PQ}+\overset{\frown}{QR}+\overset{\frown}{RS}+\overset{\frown}{ST}+\overset{\frown}{TU}+\overset{\frown}{UP}$
$m\overset{\frown}{PQS}=90^\circ+90^\circ+50^\circ+40^\circ+50^\circ+40^\circ=360^\circ$

Step10: Calculate $m\overset{\frown}{PRU}$ (19)

Add $\overset{\frown}{PQ}+\overset{\frown}{QR}+\overset{\frown}{RS}+\overset{\frown}{ST}+\overset{\frown}{TU}$
$m\overset{\frown}{PRU}=90^\circ+90^\circ+50^\circ+40^\circ+50^\circ=320^\circ$

Answer:

1. $230^\circ$
2. $180^\circ$
3. $90^\circ$
4. $50^\circ$
5. $140^\circ$
6. $130^\circ$
7. $360^\circ$
8. $320^\circ$