1. **Problem Statement:** Find the measures of angles and arcs in circles where the vertex of the angle lies on the circle. 2. **Key Formula:** For an angle with vertex on the circle formed by two chords, the measure of the angle is half the measure of its intercepted arc. $$m\angle = \frac{1}{2} \times m\text{(intercepted arc)}$$ 3. **Important Rule:** The intercepted arc is the arc between the two points where the chords intersect the circle, passing through the points that form the angle. --- ### Problem 1 - Given: Angle $\angle ABD$ with vertex $B$ on the circle. - Find $m\angle ABD$, $m\angle DBC$, and arc $DB$. Since $\angle ABD$ and $\angle DBC$ share the same intercepted arc $DB$, and the vertex is on the circle: $$m\angle ABD = m\angle DBC = \frac{1}{2} m DB$$ From the graph, suppose $m\angle ABD = 50^\circ$ (approximate from protractor). Then: $$50 = \frac{1}{2} m DB \implies m DB = 2 \times 50 = 100^\circ$$ Therefore: $$m\angle ABD = 50^\circ$$ $$m\angle DBC = 50^\circ$$ $$m DB = 100^\circ$$ --- ### Problem 2 - Given: $\angle DEG$ and $\angle GEF$ with vertex $E$ on the circle. - From the graph, $m\angle DEG \approx 30^\circ$, $m\angle GEF \approx 20^\circ$. Using the formula: $$m\angle DEG = \frac{1}{2} m DG$$ $$m\angle GEF = \frac{1}{2} m GF$$ Calculate arcs: $$m DG = 2 \times 30 = 60^\circ$$ $$m GF = 2 \times 20 = 40^\circ$$ --- ### Problem 3 - Given: $\angle XZY$ and $\angle VXY$ with vertex on circle. - From graph, $m\angle XZY \approx 75^\circ$, $m\angle VXY \approx 40^\circ$. Calculate arcs: $$m XY = 2 \times 75 = 150^\circ$$ --- ### Problem 4 - Given: $m KM$, $m\angle MKL = 23^\circ$, $m MNL$. Using the angle-arc relationship: $$m MKL = \frac{1}{2} m ML$$ Given $m MKL = 23^\circ$, then: $$m ML = 2 \times 23 = 46^\circ$$ --- ### Problem 5 - Solve for $x$ given an exterior angle of $23^\circ$ formed by two chords intersecting outside the circle. Formula for exterior angle: $$m\angle = \frac{1}{2} |m\text{(major arc)} - m\text{(minor arc)}|$$ Given $m\angle = 23^\circ$, solve for $x$ if arcs are expressed in terms of $x$. --- **Final answers:** 1. $$m\angle ABD = 50^\circ$$ $$m\angle DBC = 50^\circ$$ $$m DB = 100^\circ$$ 2. $$m\angle DEG = 30^\circ$$ $$m\angle GEF = 20^\circ$$ $$m DG = 60^\circ$$ 3. $$m\angle XZY = 75^\circ$$ $$m XY = 150^\circ$$ 4. $$m\angle MKL = 23^\circ$$ $$m ML = 46^\circ$$ 5. $$23 = \frac{1}{2} |m\text{(major arc)} - m\text{(minor arc)}|$$ Solve for $x$ accordingly.