In the figure shown, segment DC in triangle ACD is extended to point B. Which of the following angles is an obtuse angle?

Rationale

∠ACD is an obtuse angle.

In triangle ACD, extending segment DC to point B results in angle ∠ACD being greater than 90 degrees. This makes ∠ACD the only angle among the provided options that is classified as obtuse, as it exceeds the right angle threshold.

A (∠CAD) YOUR ANSWER

Angle ∠CAD is formed by segment AC and segment AD. Since triangle ACD consists of interior angles that sum to 180 degrees, and considering that angle ∠ACD is obtuse, angle ∠CAD must be acute and therefore cannot be obtuse.

B (∠ACD) CORRECT

Angle ∠ACD is formed at vertex C between the segments AC and CD. As stated, extending segment DC to point B makes angle ∠ACD greater than 90 degrees, categorizing it as an obtuse angle. This angle is the only one in the options that possesses this property.

C (∠ADC) YOUR ANSWER

Angle ∠ADC is located at vertex D between segments AD and DC. Since it is an interior angle of triangle ACD, and the sum of the angles in a triangle is always 180 degrees, ∠ADC, along with the other angles, must be acute to accommodate the obtuse angle ∠ACD.

D (∠ACB) YOUR ANSWER

Angle ∠ACB is an exterior angle formed by extending segment DC. However, this angle is supplementary to angle ∠ACD and does not exceed 180 degrees. Given that it is adjacent to an obtuse angle, it remains acute and cannot be classified as obtuse.

Conclusion

In triangle ACD, the extension of segment DC to point B causes angle ∠ACD to be the only obtuse angle among the options provided. The other angles, ∠CAD, ∠ADC, and ∠ACB, are either acute or supplementary to the obtuse angle and, therefore, do not meet the criteria for being classified as obtuse. This understanding of angle classification is essential in geometric analysis and problem-solving.

In the figure shown, segment DC in triangle ACD is extended to point B. Which of the following angles is an obtuse angle?

∠ACD is an obtuse angle.

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