3. answer the following for the given figure. (lesson 4.2) (1 point)
a. suppose $\angle 1$ is $105^{\circ}$. list all the angles congruent to $\angle 1$.
b. suppose $\angle 3$ is also $105^{\circ}$. name all angles supplementary to $\angle 3$. what is their measurement?

4. line $r$ and line $s$ are parallel. answer the following about the given figure. (lesson 4.2) (1 point)
c. which angle pairs are alternate interior angles? list all angle pairs.
d. what theorem explains why $\angle 4 \cong \angle 8$?
e. which angle pairs are same-side interior angles? list all angle pairs.

Question

3. answer the following for the given figure. (lesson 4.2) (1 point)
a. suppose $\angle 1$ is $105^{\circ}$. list all the angles congruent to $\angle 1$.
b. suppose $\angle 3$ is also $105^{\circ}$. name all angles supplementary to $\angle 3$. what is their measurement?

4. line $r$ and line $s$ are parallel. answer the following about the given figure. (lesson 4.2) (1 point)
c. which angle pairs are alternate interior angles? list all angle pairs.
d. what theorem explains why $\angle 4 \cong \angle 8$?
e. which angle pairs are same-side interior angles? list all angle pairs.

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vertical Angles", "Alternate Interior Angles", "Corresponding Angles Postulate", "Supplementary Angles", "Same-Side Interior Angles" ], "new_concepts": [], "current_concepts": [ "Vertical Angles", "Alternate Interior Angles", "Corresponding Angles Postulate", "Supplementary Angles", "Same-Side Interior Angles" ] } ## Identify supplementary angles to $\angle 3$ Using the *Supplementary Angles* knowledge point $$ \begin{aligned} &\angle 3 = 105^\circ\ &\text{Linear pairs with } \angle 3: \angle 4 \text{ and } \angle 6\ &\text{Consecutive interior with } \angle 3: \angle 2\ &\text{Consecutive exterior with } \angle 3 \text{ (via parallel lines): } \angle 8\ &\text{Supplementary angles: } \angle 2, \angle 4, \angle 6, \angle 8\ &\text{Measurement: } 180^\circ - 105^\circ = 75^\circ \end{aligned} $$ ## Identify alternate interior angle pairs Using the *Alternate Interior Angles* knowledge point $$ \begin{aligned} &\text{Interior angles between parallel lines } r \text{ and } s: \angle 3, \angle 4, \angle 5, \angle 6\ &\text{Alternate sides of transversal } t:\ &\text{Pair 1: } \angle 3 \text{ and } \angle 5\ &\text{Pair 2: } \angle 4 \text{ and } \angle 6 \end{aligned} $$ ## Determine the theorem for $\angle 4 \cong \angle 8$ Using the *Corresponding Angles Postulate* and *Vertical Angles* knowledge points $$ \begin{aligned} &\angle 4 \text{ and } \angle 8 \text{ lie in the same relative position at each intersection.}\ &\text{Theorem: Corresponding Angles Postulate} \end{aligned} $$ ## Identify same-side interior angle pairs Using the *Same-Side Interior Angles* knowledge point $$ \begin{aligned} &\text{Interior angles on the same side of transversal } t:\ &\text{Pair 1: } \angle 3 \text{ and } \angle 6\ &\text{Pair 2: } \angle 4 \text{ and } \angle 5 \end{aligned} $$ ## Question 3b The angles supplementary to $\angle 3$ are $\angle 2$, $\angle 4$, $\angle 6$, and $\angle 8$. Their measurement is $75^\circ$. ## Question 4c The alternate interior angle pairs are: - $\angle 3$ and $\angle 5$ - $\angle 4$ and $\angle 6$ ## Question 4d The Corresponding Angles Postulate. ## Question 4e The same-side interior angle pairs are: - $\angle 3$ and $\angle 6$ - $\angle 4$ and $\angle 5$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Line Angles" ] }

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QUESTION IMAGE

Question

Explanation:

Identify supplementary angles to \(\angle 3\)

Identify alternate interior angle pairs

Determine the theorem for \(\angle 4 \cong \angle 8\)

Identify same-side interior angle pairs

Answer:

Question 3b

Question 4c

Question 4d

Question 4e