question 16 of 25

in the figure below, $\angle abd$ measures $41^\circ$. what is the measure of $\angle abc$?

a. $51^\circ$
b. $82^\circ$
c. $41^\circ$
d. $72^\circ$

Question

question 16 of 25

in the figure below, $\angle abd$ measures $41^\circ$. what is the measure of $\angle abc$?

a. $51^\circ$
b. $82^\circ$
c. $41^\circ$
d. $72^\circ$

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": [ "HL Congruence Theorem", "Angle Bisector Theorem" ], "new_concepts": [], "current_concepts": [ "HL Congruence Theorem", "Angle Bisector Theorem", "Right Triangle Congruence Theorems" ] } ## Identify the given geometric properties Using the **HL Congruence Theorem** and **Angle Bisector Theorem** knowledge points $$ \begin{aligned} &\angle BCD = 90^\circ, \quad \angle BAD = 90^\circ\ &CD = AD\ &BD = BD \quad \text{(Reflexive Property)} \end{aligned} $$ ## Prove triangle congruence Using the **HL Congruence Theorem** knowledge point $$ \begin{aligned} &\triangle BCD \cong \triangle BAD \quad \text{(Hypotenuse-Leg Congruence)} \end{aligned} $$ ## Find the corresponding angle measure Using the **HL Congruence Theorem** knowledge point $$ \begin{aligned} &\angle CBD = \angle ABD\ &\angle CBD = 41^\circ \end{aligned} $$ ## Calculate the total angle measure Using the **Angle Bisector Theorem** knowledge point $$ \begin{aligned} &\angle ABC = \angle CBD + \angle ABD\ &\angle ABC = 41^\circ + 41^\circ = 82^\circ \end{aligned} $$

A. $51^\circ$

B. $82^\circ$

C. $41^\circ$

D. $72^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "HL Congruence Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given geometric properties

Prove triangle congruence

Find the corresponding angle measure

Calculate the total angle measure

Answer: