if $abcd$ is a rectangle and $m\angle cbd = 47^{\circ}$, what is the value of $x$?

a. 47
b. 133
c. 94
d. 86
e. 43
f. cannot be determined

Question

if $abcd$ is a rectangle and $m\angle cbd = 47^{\circ}$, what is the value of $x$?

a. 47
b. 133
c. 94
d. 86
e. 43
f. cannot be determined

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": [ "Properties of Rectangles", "Isosceles Triangle Theorem", "Triangle Angle Sum Theorem" ], "new_concepts": [], "current_concepts": [ "Properties of Rectangles", "Isosceles Triangle Theorem", "Triangle Angle Sum Theorem" ] } ## Identify properties of the diagonals Using the *Properties of Rectangles* knowledge point $$ \begin{aligned} &AC = BD \quad \text{(Diagonals are equal in length)}\ &OA = OB = OC = OD \quad \text{(Diagonals bisect each other)} \end{aligned} $$ where $O$ is the intersection point of diagonals $AC$ and $BD$. ## Apply properties to triangle OBC Using the *Isosceles Triangle Theorem* knowledge point $$ \begin{aligned} &OB = OC \implies \triangle OBC \text{ is isosceles}\ &m\angle OCB = m\angle OBC = 47^\circ \end{aligned} $$ ## Calculate the angle at the intersection Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &m\angle BOC + m\angle OBC + m\angle OCB = 180^\circ\ &m\angle BOC + 47^\circ + 47^\circ = 180^\circ\ &m\angle BOC = 180^\circ - 94^\circ = 86^\circ \end{aligned} $$ ## Solve for the target angle x Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &x + m\angle BOC = 180^\circ \quad \text{(Linear pair)}\ &x = 180^\circ - 86^\circ = 94^\circ \end{aligned} $$

(A) 47

(B) 133

(C) 94

(D) 86

(E) 43

(F) Cannot be determined

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Properties of Rectangles" ] }

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QUESTION IMAGE

Question

Explanation:

Identify properties of the diagonals

Apply properties to triangle OBC

Calculate the angle at the intersection

Solve for the target angle x

Answer: