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\angle A \text{ intercepts } \widehat{BC}, \angle D \text{ intercepts } \widehat{BC}

2.		m\angle A = \frac{1}{2} m\widehat{BC}2.?

3.		m\angle D = \frac{1}{2} m\widehat{BC}3.?

4.		m\angle A = m\angle D4.\text{substitution property}

5.		\angle A \cong \angle D5.\text{definition of congruent angles}






Which reason correctly fills in the missing steps 2 and 3?


Ainscribed angles theorem
Bthird corollary to the inscribed angles theorem
Ccentral angle of a triangle has the same measure as its intercepted arc.
D\angle formed by a tangent and a chord is half the measure of the intercepted arc.

Verified Answer :

  • The "inscribed angles theorem" states that the measure of an inscribed angle is half the measure of its intercepted arc.
  • Step 2 claims that the measure of \(\angle A\) is half the measure of arc \(\overparen{BC}\), which corresponds directly to the definition provided by the inscribed angles theorem.
  • Step 3 makes an identical claim for \(\angle D\), justifying it with the same inscribed angles theorem.
  • Final answer: Both step 2 and step 3 are justified by the \(\boxed{\text{inscribed angles theorem}}\).

Explanation

  1. Analysis of the inscribed angles theorem and the proof steps.

The problem presents a two-column proof aimed at demonstrating that two angles in a circle, \(\angle A\) and \(\angle D\), are equal because they intercept the same arc, \(\overparen{BC}\). The steps require justifications for the claim that the measure of these inscribed angles is half the measure of their intercepted arc. The "inscribed angles theorem" states that the measure of an inscribed angle in a circle is exactly half the measure of its intercepted arc. Since both \(\angle A\) and \(\angle D\) are inscribed angles intercepting the same arc \(\overparen{BC}\), this theorem directly justifies both step 2 and step 3.

  1. Assigning the correct justification to the proof steps.

The "inscribed angles theorem" is the standard geometric principle establishing that an inscribed angle's measure is one-half of the intercepted arc's measure. Applying this to step 2: \(m\angle A = \frac{1}{2} m\overparen{BC}\), the justification is the "inscribed angles theorem". Similarly, for step 3: \(m\angle D = \frac{1}{2} m\overparen{BC}\), the justification is also the "inscribed angles theorem". Both steps follow the same fundamental property of circles.

  1. Conclusion of the proof analysis.

The identification of the justifications is complete. The "inscribed angles theorem" applies to both step 2 and step 3. The final answer for both boxes is the "inscribed angles theorem".

Examples

This theorem is fundamental in geometry, such as when designing a circular window or a radar system. If you know the measure of the arc covered by an object along the edge of a circle, the inscribed angles theorem allows you to calculate the viewing angle from any point on the opposite side of the circle's circumference. For instance, if an arc \(\overparen{BC}\) measures \(80^\circ\), then any angle \(\angle A\) subtending that arc will measure \(\frac{80^\circ}{2} = 40^\circ\). This is a crucial concept for engineers and architects working with circular structures.

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