Question

1. find the measure of $\angle abc$.
2. find the measure of $\angle lmn$.
3. find the measure of $\overarc{bc}$.
4. find the measure of $\overarc{xwz}$.
5. if the $m\angle acb = 80^\circ$, then find $m\angle bda$.
6. given that $\overline{ab}$ is a diameter, find the measure of $x^\circ$ and $y^\circ$.

Explanation:

19. Step1: Use inscribed arc-angle theorem

An inscribed angle is half its intercepted arc. $\angle ABC$ intercepts $\overset{\frown}{AC}=110^\circ$, so:
$\angle ABC = \frac{1}{2} \times 110^\circ$

19. Step2: Calculate the angle

$\angle ABC = 55^\circ$

20. Step1: Use inscribed arc-angle theorem

$\angle LMN$ intercepts $\overset{\frown}{LN}=68^\circ$, so:
$\angle LMN = \frac{1}{2} \times 68^\circ$

20. Step2: Calculate the angle

$\angle LMN = 34^\circ$

21. Step1: Use inscribed arc-angle theorem

The inscribed angle $\angle CAB=32^\circ$ intercepts $\overset{\frown}{BC}$, so:
$\overset{\frown}{BC} = 2 \times 32^\circ$

21. Step2: Calculate the arc measure

$\overset{\frown}{BC} = 64^\circ$

22. Step1: Use inscribed arc-angle theorem

The inscribed angle $\angle XYZ=103^\circ$ intercepts the major arc $\overset{\frown}{XWZ}$. The measure of a major arc intercepted by an inscribed angle is $2 \times$ the angle, so:
$\overset{\frown}{XWZ} = 2 \times 103^\circ$

22. Step2: Calculate the arc measure

$\overset{\frown}{XWZ} = 206^\circ$

23. Step1: Relate central and inscribed angles

$\angle ACB$ is a central angle intercepting $\overset{\frown}{AB}$, $\angle BDA$ is an inscribed angle intercepting the same arc, so:
$\angle BDA = \frac{1}{2} \times \angle ACB$

23. Step2: Substitute and calculate

$\angle BDA = \frac{1}{2} \times 80^\circ = 40^\circ$

24. Step1: Find $x^\circ$ (right angle theorem)

Since $\overline{AB}$ is a diameter, $\angle ACB$ is an inscribed angle intercepting a semicircle, so:
$x^\circ = 90^\circ$

24. Step2: Find $y^\circ$ (triangle angle sum)

In $\triangle ABC$, sum of angles is $180^\circ$, so:
$y^\circ = 180^\circ - 90^\circ - 50^\circ$

24. Step3: Calculate $y^\circ$

$y^\circ = 40^\circ$

Answer:

1. $\boldsymbol{55^\circ}$
2. $\boldsymbol{34^\circ}$
3. $\boldsymbol{64^\circ}$
4. $\boldsymbol{206^\circ}$
5. $\boldsymbol{40^\circ}$
6. $\boldsymbol{x^\circ=90^\circ}$, $\boldsymbol{y^\circ=40^\circ}$