if $abcd$ is a rectangle, and $m\angle adb = 55^\circ$, what is the value of $x$?

a. 80
b. 90
c. 40
d. 70
e. 110
f. cannot be determined

Question

if $abcd$ is a rectangle, and $m\angle adb = 55^\circ$, what is the value of $x$?

a. 80
b. 90
c. 40
d. 70
e. 110
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": [ "Triangle Angle Sum Theorem" ], "new_concepts": [ "Rectangle Diagonals Theorem", "Properties of Rectangles" ], "current_concepts": [ "Triangle Angle Sum Theorem", "Rectangle Diagonals Theorem", "Properties of Rectangles" ] } ## Identify properties of the rectangle Let the intersection of the diagonals $AC$ and $BD$ be point $P$. In rectangle $ABCD$, the diagonals are congruent and bisect each other. This means: $$ AP = BP = CP = DP $$ Thus, triangle $APD$ is an isosceles triangle with $AP = DP$. ## Determine the base angles of the isosceles triangle Since $AP = DP$, the angles opposite these sides are equal: $$ m\angle DAP = m\angle ADB = 55^\circ $$ ## Calculate the vertex angle Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &m\angle APD + m\angle DAP + m\angle ADB = 180^\circ\ &m\angle APD + 55^\circ + 55^\circ = 180^\circ\ &m\angle APD + 110^\circ = 180^\circ\ &m\angle APD = 70^\circ \end{aligned} $$ ## Find the value of x The angle labeled $x^\circ$ and $\angle APD$ are vertical angles at the intersection of the diagonals: $$ x = m\angle APD = 70 $$

A. 80

B. 90

C. 40

D. 70

E. 110

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 rectangle

Determine the base angles of the isosceles triangle

Calculate the vertex angle

Find the value of x

Answer: