Finding an angle from intercepted arcs in a circle

Expert intro: This problem uses the circle angle theorem for an inscribed angle formed by two chords. The key is to identify the intercepted arcs and combine them correctly.

Detailed walkthrough

Key idea: inscribed angle and intercepted arcs

An angle formed by two chords inside a circle has a measure equal to half the sum of the measures of its intercepted arcs. In this problem, $\angle DAC$ is determined by the arcs $\overset{\frown}{HL}$ and $\overset{\frown}{DC}$, so the arc measures are the main data to use.

For an inscribed angle, the formula is:

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\$$ ## Step-by-step calculation Substitute the given values: \ $$m\angle DAC = \frac{1}{2}(114^\circ + 84^\circ) \$$ \ $$m\angle DAC = \frac{1}{2}(198^\circ) \$$ \ $$m\angle DAC = 99^\circ \$$ So the measure of \$\angle DAC\$ is \$99^\circ\$. The most important reading skill here is recognizing that the question is not asking for an arc measure directly. It is asking for an angle measure, so the answer must be half of the relevant arc sum. ## Common geometry facts to remember - An inscribed angle equals half its intercepted arc measure. - If two arcs are given, check whether the angle intercepts both arcs or only one. - Always match the named points on the angle with the circle diagram before substituting values. ## Final answer \ $$\boxed{99^\circ}\$$ ### 💡 Pitfall guide A common mistake is adding the two arcs and stopping at 198° without taking half. That would be the total intercepted arc measure, not the angle measure. Another frequent error is using the wrong arc pair because the point labels can be easy to mix up when the diagram is not redrawn carefully. Before computing, confirm that the angle is inscribed and that the arcs listed are exactly the ones intercepted by its sides. ### 🔄 Real-world variant If the problem changed to ask for an angle with the same intercepted arcs but a different vertex on the circle, the method would still be the same: use half the sum of the intercepted arcs. For example, if the question were, “If \$\overset{\frown}{HL}=114^\circ\$ and \$\overset{\frown}{DC}=84^\circ\$, find \$m\angle ABC\$,” and \$\angle ABC\$ intercepted those same arcs, the result would still be \$99^\circ\$. If the vertex moved to the center of the circle, however, the angle would equal the full intercepted arc sum instead of half. ### 🔍 Related terms intercepted arcs, inscribed angle theorem, circle angle measure