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

$45^\circ$
$65^\circ$
$115^\circ$
$135^\circ$

Question

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

$45^\circ$
$65^\circ$
$115^\circ$
$135^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Trapezoid Properties", "Isosceles Trapezoid Base Angles Theorem" ], "new_concepts": [], "current_concepts": [ "Trapezoid Properties", "Isosceles Trapezoid Base Angles Theorem" ] } ## 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: $$ m\angle ADC = m\angle BAD = 115^\circ $$

(A) $45^\circ$

(B) $65^\circ$

(C) $115^\circ$

(D) $135^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Isosceles Trapezoid Base Angles Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the quadrilateral type

Apply base angle properties

Calculate the target angle

Answer: