1. a. Find $w$ given arcs 110° and 70° in a circle with inscribed angle $w$. - The inscribed angle theorem states: an inscribed angle equals half the measure of its intercepted arc. - The intercepted arc is $110° + 70° = 180°$. - So, $w = \frac{180°}{2} = 90°$. 1. b. Find $y$ given an exterior angle 30°, an arc 110°, and angle $y$ inside the circle. - Exterior angle theorem for circles: exterior angle equals half the difference of the intercepted arcs. - Here, $y$ is the inscribed angle intercepting the 110° arc. - The exterior angle 30° equals half the difference between arcs intercepted by $y$ and the other arc. - Let the other arc be $x$, then $30° = \frac{|110° - x|}{2}$. - Solving: $|110° - x| = 60°$. - So, $x = 110° - 60° = 50°$ or $x = 110° + 60° = 170°$. - Since $y$ intercepts arc 110°, $y = \frac{110°}{2} = 55°$. 1. c. Find $x$ given inscribed angle 35°, arc 30°, and angle $x$ inside the circle. - Inscribed angle theorem: angle equals half its intercepted arc. - If 35° is inscribed angle intercepting arc $A$, then $A = 2 \times 35° = 70°$. - Given arc 30°, $x$ intercepts the remaining arc: $360° - 70° - 30° = 260°$. - So, $x = \frac{260°}{2} = 130°$. 2. a. Find $x$ in triangle with sides 20, 14, 16, and $x$. - If chords form a triangle, use the triangle inequality or Pythagorean theorem if right triangle. - Assuming $x$ is the missing side opposite the largest side 20. - Using Ptolemy's theorem or chord properties is complex; assuming $x$ is the fourth side in cyclic quadrilateral. - Using Ptolemy's theorem: $20 \times x = 14 \times 16$. - So, $20x = 224$. - Cancel common factor: $\cancel{20}x = \frac{224}{\cancel{20}}$. - $x = \frac{224}{20} = 11.2$. 2. b. Find $m$ in circle divided by perpendicular chords with segments 6.5, 3, 7, and $m$. - Intersecting chords theorem: product of segments of one chord equals product of segments of the other. - So, $6.5 \times m = 3 \times 7$. - $6.5m = 21$. - Cancel common factor: $\cancel{6.5}m = \frac{21}{\cancel{6.5}}$. - $m = \frac{21}{6.5} = 3.23$ (approx). Final answers: - $w = 90°$ - $y = 55°$ - $x = 130°$ - $x = 11.2$ - $m = 3.23$