lines cd and ce are tangent to circle a. if $m\angle dae = 140^\circ$, what is the measure of $\angle dce$? (1 point)

$80^\circ$
$140^\circ$
$40^\circ$
$220^\circ$

Question

lines cd and ce are tangent to circle a. if $m\angle dae = 140^\circ$, what is the measure of $\angle dce$? (1 point)

$80^\circ$
$140^\circ$
$40^\circ$
$220^\circ$

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": [ "Circle Geometry" ], "new_concepts": [ "Tangent-Tangent Angle Theorem", "Quadrilateral Angle Sum" ], "current_concepts": [ "Circle Geometry", "Tangent-Tangent Angle Theorem", "Quadrilateral Angle Sum" ] } ## Identify properties of tangent lines to a circle Using the *Circle Geometry* knowledge point $$ \begin{aligned} &AD \perp CD \implies m\angle ADC = 90^\circ\ &AE \perp CE \implies m\angle AEC = 90^\circ \end{aligned} $$ ## Calculate the measure of angle DCE Using the *Circle Geometry* knowledge point $$ \begin{aligned} &m\angle DCE + m\angle ADC + m\angle AEC + m\angle DAE = 360^\circ\ &m\angle DCE + 90^\circ + 90^\circ + 140^\circ = 360^\circ\ &m\angle DCE + 320^\circ = 360^\circ\ &m\angle DCE = 40^\circ \end{aligned} $$

(A) $80^\circ$

(B) $140^\circ$

(C) $40^\circ$

(D) $220^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Geometry" ] }

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QUESTION IMAGE

Question

Explanation:

Identify properties of tangent lines to a circle

Calculate the measure of angle DCE

Answer: