your turn
1. in $\delta jkl$, $m\angle j = 40^{\circ}$ and $m\angle k = 55^{\circ}$. in $\delta mnp$, $m\angle m = 40^{\circ}$ and $m\angle p = 85^{\circ}$. a student concludes that the triangles are not similar. do you agree or disagree? why?

2. in the figure, $\overline{ab} \parallel \overline{de}$. prove that $\delta acb \sim \delta ecd$. use the statement & reason bank to help you complete the proof.

statements | reasons
1. | 1. given
2. | 2. vertical angles are congruent
3. | 3. alternate interior angles are congruent
4. $\delta acb \sim \delta ecd$ | 4.

statement & reason bank:
$\angle 1 \cong \angle 4$
aa triangle similarity theorem
$\overline{ab} \parallel \overline{de}$
$\angle 2 \cong \angle 3$

Question

your turn
1. in $\delta jkl$, $m\angle j = 40^{\circ}$ and $m\angle k = 55^{\circ}$. in $\delta mnp$, $m\angle m = 40^{\circ}$ and $m\angle p = 85^{\circ}$. a student concludes that the triangles are not similar. do you agree or disagree? why?

2. in the figure, $\overline{ab} \parallel \overline{de}$. prove that $\delta acb \sim \delta ecd$. use the statement & reason bank to help you complete the proof.

statement & reason bank:
$\angle 1 \cong \angle 4$
aa triangle similarity theorem
$\overline{ab} \parallel \overline{de}$
$\angle 2 \cong \angle 3$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Triangle Angle-Sum Theorem", "Vertical Angles", "Alternate Interior Angles" ], "new_concepts": [ "AA Triangle Similarity Theorem" ], "current_concepts": [ "Triangle Angle-Sum Theorem", "Vertical Angles", "Alternate Interior Angles", "AA Triangle Similarity Theorem" ] } ## Calculate missing angles in Question 1 Using the *Triangle Angle-Sum Theorem* knowledge point $$ \begin{aligned} &\text{In }\Delta JKL:\quad m\angle L = 180^\circ - (40^\circ + 55^\circ) = 85^\circ\ &\text{In }\Delta MNP:\quad m\angle N = 180^\circ - (40^\circ + 85^\circ) = 55^\circ \end{aligned} $$ ## Compare corresponding angles for similarity Using the *AA Triangle Similarity Theorem* knowledge point $$ \begin{aligned} &m\angle J = m\angle M = 40^\circ\ &m\angle K = m\angle N = 55^\circ\ &m\angle L = m\angle P = 85^\circ \end{aligned} $$ Since two pairs of corresponding angles are congruent, the triangles are similar by AA similarity. Therefore, we disagree with the student's conclusion. ## Complete Statement 1 of the proof Using the *Parallel Lines* knowledge point $$ \text{Statement 1: } \overline{AB} \parallel \overline{DE} $$ ## Complete Statement 2 of the proof Using the *Vertical Angles* knowledge point $$ \text{Statement 2: } \angle 2 \cong \angle 3 $$ ## Complete Statement 3 and Reason 4 of the proof Using the *Alternate Interior Angles* and *AA Triangle Similarity Theorem* knowledge points $$ \begin{aligned} &\text{Statement 3: } \angle 1 \cong \angle 4\ &\text{Reason 4: AA Triangle Similarity Theorem} \end{aligned} $$ ## Question 1 Disagree. By calculating the third angle in each triangle using the Triangle Angle-Sum Theorem: * In $\Delta JKL$, $m\angle L = 180^\circ - (40^\circ + 55^\circ) = 85^\circ$. * In $\Delta MNP$, $m\angle N = 180^\circ - (40^\circ + 85^\circ) = 55^\circ$. Both triangles have angles measuring $40^\circ$, $55^\circ$, and $85^\circ$. Since they share two congruent angles, the triangles are similar by the AA Triangle Similarity Theorem. --- ## Question 2 The completed proof table is: | Statements | Reasons | | :--- | :--- | | 1. $\overline{AB} \parallel \overline{DE}$ | 1. Given | | 2. $\angle 2 \cong \angle 3$ | 2. Vertical Angles are Congruent | | 3. $\angle 1 \cong \angle 4$ | 3. Alternate Interior Angles are Congruent | | 4. $\Delta ACB \sim \Delta ECD$ | 4. AA Triangle Similarity Theorem | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "AA Triangle Similarity Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate missing angles in Question 1

Compare corresponding angles for similarity

Complete Statement 1 of the proof

Complete Statement 2 of the proof

Complete Statement 3 and Reason 4 of the proof

Answer:

Question 1

Question 2

Statements	Reasons
2. \(\angle 2 \cong \angle 3\)	2. Vertical Angles are Congruent
3. \(\angle 1 \cong \angle 4\)	3. Alternate Interior Angles are Congruent
4. \(\Delta ACB \sim \Delta ECD\)	4. AA Triangle Similarity Theorem