Question

what is the measure of \\(\angle adc\\) in quadrilateral abcd?

\\(45^\circ\\)
\\(65^\circ\\)
\\(115^\circ\\)
\\(135^\circ\\)

Explanation:

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<reasoning>

Identify the quadrilateral type

Using the Trapezoid Properties knowledge point

• The arrows on segments \(BC\) and \(AD\) indicate that \(BC \parallel AD\).
• The tick marks on segments \(AB\) and \(CD\) indicate that \(AB = CD\).
• A quadrilateral with one pair of parallel sides and congruent non-parallel sides is an isosceles trapezoid.

Apply base angle properties

Using the Isosceles Trapezoid Base Angles Theorem knowledge point

• In an isosceles trapezoid, the base angles sharing a parallel side are congruent.
• Here, the bases are \(BC\) and \(AD\).
• Therefore, the base angles at base \(AD\) are congruent: \(\angle BAD \cong \angle ADC\).

Calculate the target angle

Using the Isosceles Trapezoid Base Angles Theorem knowledge point

• Given \(m\angle BAD = 115^\circ\).
• Since \(\angle ADC \cong \angle BAD\), we have:

\[
m\angle ADC = m\angle BAD = 115^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(45^\circ\)</mcq-option>
<mcq-option>(B) \(65^\circ\)</mcq-option>
<mcq-correct>(C) \(115^\circ\)</mcq-correct>
<mcq-option>(D) \(135^\circ\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the quadrilateral type

Using the Trapezoid Properties knowledge point

• The arrows on segments \(BC\) and \(AD\) indicate that \(BC \parallel AD\).
• The tick marks on segments \(AB\) and \(CD\) indicate that \(AB = CD\).
• A quadrilateral with one pair of parallel sides and congruent non-parallel sides is an isosceles trapezoid.

Apply base angle properties

Using the Isosceles Trapezoid Base Angles Theorem knowledge point

• In an isosceles trapezoid, the base angles sharing a parallel side are congruent.
• Here, the bases are \(BC\) and \(AD\).
• Therefore, the base angles at base \(AD\) are congruent: \(\angle BAD \cong \angle ADC\).

Calculate the target angle

Using the Isosceles Trapezoid Base Angles Theorem knowledge point

• Given \(m\angle BAD = 115^\circ\).
• Since \(\angle ADC \cong \angle BAD\), we have:

\[
m\angle ADC = m\angle BAD = 115^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(45^\circ\)</mcq-option>
<mcq-option>(B) \(65^\circ\)</mcq-option>
<mcq-correct>(C) \(115^\circ\)</mcq-correct>
<mcq-option>(D) \(135^\circ\)</mcq-option>
</answer>

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