4. prove that lines ab and cd are parallel.

statement | reason

5. given that ab || cd and \angle 1 = \angle 2, prove ac = ab.

Question

4. prove that lines ab and cd are parallel.

statement | reason

5. given that ab || cd and \angle 1 = \angle 2, prove ac = ab.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Straight Angle Theorem", "Consecutive Interior Angles Converse" ], "new_concepts": [ "Two-Column Proof" ], "current_concepts": [ "Straight Angle Theorem", "Consecutive Interior Angles Converse", "Two-Column Proof" ] } ## Identify the given geometric values Using the *Straight Angle Theorem* knowledge point $$ \begin{aligned} &\text{Let } E \text{ and } F \text{ be the intersection points on lines } AB \text{ and } CD \text{ respectively.}\ &m\angle AEF = 148^\circ\ &m\angle EFD = 32^\circ \end{aligned} $$ ## Calculate the consecutive interior angle sum Using the *Straight Angle Theorem* knowledge point $$ \begin{aligned} &m\angle AEF + m\angle EFD = 148^\circ + 32^\circ = 180^\circ \end{aligned} $$ ## Prove lines are parallel Using the *Consecutive Interior Angles Converse* knowledge point $$ \begin{aligned} &\text{Since } m\angle AEF + m\angle EFD = 180^\circ \text{, the consecutive interior angles are supplementary.}\ &\therefore AB \parallel CD \end{aligned} $$ To prove that lines $AB$ and $CD$ are parallel, we can complete the two-column proof as follows: | Statement | Reason | | :--- | :--- | | 1. $m\angle AEF = 148^\circ$ and $m\angle EFD = 32^\circ$ | 1. Given | | 2. $m\angle AEF + m\angle EFD = 148^\circ + 32^\circ = 180^\circ$ | 2. Addition Property of Equality | | 3. $\angle AEF$ and $\angle EFD$ are supplementary | 3. Definition of supplementary angles | | 4. $AB \parallel CD$ | 4. Consecutive Interior Angles Converse | { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Consecutive Interior Angles Converse" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given geometric values

Calculate the consecutive interior angle sum

Prove lines are parallel

Answer:

Statement	Reason
2. \(m\angle AEF + m\angle EFD = 148^\circ + 32^\circ = 180^\circ\)	2. Addition Property of Equality
3. \(\angle AEF\) and \(\angle EFD\) are supplementary	3. Definition of supplementary angles
4. \(AB \parallel CD\)	4. Consecutive Interior Angles Converse