what is the measure of $\angle abc$ in the figure below?

a. $15^\circ$
b. $30^\circ$
c. $45^\circ$
d. $90^\circ$
e. $15^\circ$
f. cannot be determined

Question

what is the measure of $\angle abc$ in the figure below?

a. $15^\circ$
b. $30^\circ$
c. $45^\circ$
d. $90^\circ$
e. $15^\circ$
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": [ "Distance from Point to Line", "Pythagorean Theorem" ], "new_concepts": [ "Angle Bisector Theorem" ], "current_concepts": [ "Distance from Point to Line", "Pythagorean Theorem", "Angle Bisector Theorem" ] } ## Identify given geometric properties Using the *Distance from Point to Line* knowledge point $$ \begin{aligned} &DA \perp BA \implies d(D, \overleftrightarrow{BA}) = DA = 15\ &DC \perp BC \implies d(D, \overleftrightarrow{BC}) = DC\ &BA = BC = 15 \end{aligned} $$ ## Determine congruence of right triangles Using the *Pythagorean Theorem* knowledge point $$ \begin{aligned} &\triangle DAB \text{ and } \triangle DCB \text{ are right triangles}\ &BD \text{ is the common hypotenuse}\ &BA = BC = 15 \implies \triangle DAB \cong \triangle DCB \text{ (HL Congruence)}\ &DA = DC = 15 \end{aligned} $$ ## Calculate the angle measures Using the *Angle Bisector Theorem* knowledge point $$ \begin{aligned} &DA = DC \implies BD \text{ bisects } \angle ABC\ &\angle ABD = \angle CBD = 45^\circ\ &\angle ABC = \angle ABD + \angle CBD = 45^\circ + 45^\circ = 90^\circ \end{aligned} $$

A. $15^\circ$

B. $30^\circ$

C. $45^\circ$

D. $90^\circ$

E. $15^\circ$

F. Cannot be determined

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Angle Bisector Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given geometric properties

Determine congruence of right triangles

Calculate the angle measures

Answer: