The problem asks to find the indicated measures in three different circle diagrams. a) Find $m\angle 1$ given that $m\stackrel{\frown}{DB} = 82^{\circ}$ and $m\stackrel{\frown}{AC} = 71^{\circ}$. b) Find $m\stackrel{\frown}{UW}$ given that $m\angle V = 26^{\circ}$ and $m\angle UTV = 36^{\circ}$. c) Find $m\angle P$ given that $m\stackrel{\frown}{KL} = 52^{\circ}$ and $m\stackrel{\frown}{JN} = 18^{\circ}$.

1. Problem Description

The problem asks to find the indicated measures in three different circle diagrams.

a) Find $m\angle 1$ given that $m\stackrel{\frown}{DB} = 82^{\circ}$ and $m\stackrel{\frown}{AC} = 71^{\circ}$.

b) Find $m\stackrel{\frown}{UW}$ given that $m\angle V = 26^{\circ}$ and $m\angle UTV = 36^{\circ}$.

c) Find $m\angle P$ given that $m\stackrel{\frown}{KL} = 52^{\circ}$ and $m\stackrel{\frown}{JN} = 18^{\circ}$.

2. Solution Steps

a) To find the measure of the angle formed by two chords intersecting inside a circle, we use the formula:

$m\angle 1 = \frac{1}{2} (m\stackrel{\frown}{DB} + m\stackrel{\frown}{AC})$

$m\angle 1 = \frac{1}{2} (82^{\circ} + 71^{\circ})$

$m\angle 1 = \frac{1}{2} (153^{\circ})$

$m\angle 1 = 76.5^{\circ}$

b) We are given $m\angle V = 26^{\circ}$ and $m\angle UTV = 36^{\circ}$.

The angle $UTV$ is a central angle, so $m\stackrel{\frown}{UV} = m\angle UTV = 36^{\circ}$.

Since the entire circle is $360^{\circ}$, we have $m\stackrel{\frown}{VU} + m\stackrel{\frown}{UW} + m\stackrel{\frown}{WV} = 360^{\circ}$.

Also, since $\angle V = 26^{\circ}$ is an inscribed angle subtending the arc $\stackrel{\frown}{UW}$, we have $m\stackrel{\frown}{UW} = 2 \times m\angle V = 2 \times 26^{\circ} = 52^{\circ}$.

Therefore, $m\stackrel{\frown}{UW} = 52^{\circ}$.

c) To find the measure of an angle formed by two secants intersecting outside a circle, we use the formula:

$m\angle P = \frac{1}{2} (m\stackrel{\frown}{KL} - m\stackrel{\frown}{JN})$

$m\angle P = \frac{1}{2} (52^{\circ} - 18^{\circ})$

$m\angle P = \frac{1}{2} (34^{\circ})$

$m\angle P = 17^{\circ}$

3. Final Answer

a) $m\angle 1 = 76.5^{\circ}$

b) $m\stackrel{\frown}{UW} = 52^{\circ}$

c) $m\angle P = 17^{\circ}$