question 7 (multiple choice worth 1 points)
if $m\angle abc = 53^{\circ}$ and $m\angle ecd = 40^{\circ}$, which statement explains whether the aa similarity postulate can be used to determine whether $\delta bac \sim \delta edc$?

Question

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": [ "AA Similarity Postulate" ], "new_concepts": [ "Triangle Angle Sum Theorem" ], "current_concepts": [ "AA Similarity Postulate", "Triangle Angle Sum Theorem", "Right Triangle Properties" ] } ## Identify given angles and target triangles Using the *AA Similarity Postulate* knowledge point $$ \begin{aligned} &\triangle BAC: \angle A = 90^\circ, \quad \angle ABC = 53^\circ\ &\triangle EDC: \angle D = 90^\circ, \quad \angle ECD = 40^\circ \end{aligned} $$ ## Calculate the remaining angle in the first triangle We use the Triangle Angle Sum Theorem to find the measure of $\angle BCA$: $$ \begin{aligned} \text{m}\angle BCA &= 180^\circ - \text{m}\angle A - \text{m}\angle ABC \ \text{m}\angle BCA &= 180^\circ - 90^\circ - 53^\circ = 37^\circ \end{aligned} $$ ## Calculate the remaining angle in the second triangle We use the Triangle Angle Sum Theorem to find the measure of $\angle DEC$: $$ \begin{aligned} \text{m}\angle DEC &= 180^\circ - \text{m}\angle D - \text{m}\angle ECD \ \text{m}\angle DEC &= 180^\circ - 90^\circ - 40^\circ = 50^\circ \end{aligned} $$ ## Compare corresponding angles for similarity We compare the sets of interior angles for both triangles: $$ \begin{aligned} &\text{Angles of } \triangle BAC: \{90^\circ, 53^\circ, 37^\circ\} \ &\text{Angles of } \triangle EDC: \{90^\circ, 50^\circ, 40^\circ\} \end{aligned} $$ Since the acute angles are not equal ($53^\circ eq 50^\circ$ and $37^\circ eq 40^\circ$), the triangles do not have two pairs of congruent angles. Thus, the AA similarity postulate cannot be used because the triangles are not similar.

(A) Yes, because $\text{m}\angle BCA = 37^\circ$ and $\text{m}\angle DEC = 37^\circ$.

(B) Yes, because $\text{m}\angle BCA = 40^\circ$ and $\text{m}\angle DEC = 53^\circ$.

(C) No, because $\text{m}\angle BCA = 53^\circ$ and $\text{m}\angle DEC = 40^\circ$.

(D) No, because $\text{m}\angle BCA = 37^\circ$ and $\text{m}\angle DEC = 50^\circ$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "AA Similarity Postulate" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify given angles and target triangles

Calculate the remaining angle in the first triangle

Calculate the remaining angle in the second triangle

Compare corresponding angles for similarity

Answer: