group c - angles in parallel lines, including triangles
calculate the size of angle $\theta$. you must use the sum of angles in a triangle in each solution.
state any other angle facts that you use.

1)
$a$, $b$, $c$, $d$, $e$
$\theta$, $74^\circ$, $28^\circ$

2)
$a$, $b$, $c$
$52^\circ$, $68^\circ$, $\theta$

3)
$a$, $b$, $c$, $d$, $e$
$41^\circ$, $\theta$

4)
$a$, $b$, $c$, $d$, $e$
$32^\circ$, $116^\circ$, $\theta$

5)
$a$, $b$, $c$
$47^\circ$, $18^\circ$, $\theta$

6)
$a$, $b$, $c$, $d$, $e$
$77^\circ$, $\theta$, $36^\circ$

7)
$a$, $b$, $c$, $d$, $e$
$60^\circ$, $\theta$, $120^\circ$

8)
$a$, $b$, $c$, $d$, $e$
$135^\circ$, $\theta$

9)
$a$, $b$, $c$
$65^\circ$, $45^\circ$, $45^\circ$, $\theta$

Question

group c - angles in parallel lines, including triangles
calculate the size of angle $\theta$. you must use the sum of angles in a triangle in each solution.
state any other angle facts that you use.

1)
$a$, $b$, $c$, $d$, $e$
$\theta$, $74^\circ$, $28^\circ$

2)
$a$, $b$, $c$
$52^\circ$, $68^\circ$, $\theta$

3)
$a$, $b$, $c$, $d$, $e$
$41^\circ$, $\theta$

4)
$a$, $b$, $c$, $d$, $e$
$32^\circ$, $116^\circ$, $\theta$

5)
$a$, $b$, $c$
$47^\circ$, $18^\circ$, $\theta$

6)
$a$, $b$, $c$, $d$, $e$
$77^\circ$, $\theta$, $36^\circ$

7)
$a$, $b$, $c$, $d$, $e$
$60^\circ$, $\theta$, $120^\circ$

8)
$a$, $b$, $c$, $d$, $e$
$135^\circ$, $\theta$

9)
$a$, $b$, $c$
$65^\circ$, $45^\circ$, $45^\circ$, $\theta$

Accepted Answer

{ "quality": "clear", "question_count": 9, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Parallel Line Angles", "Transversal Lines", "Angle Relationships", "Triangle Angle Sum Theorem" ], "new_concepts": [], "current_concepts": [ "Parallel Line Angles", "Transversal Lines", "Angle Relationships", "Triangle Angle Sum Theorem" ] } ## Solve Question 1 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle ECD = 180^\circ - 74^\circ = 106^\circ\ &\angle CED = 180^\circ - (106^\circ + 28^\circ) = 46^\circ\ &\theta = \angle CED = 46^\circ \end{aligned} $$ ## Solve Question 2 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle BCA = \angle CAD = 52^\circ\ &\angle BCD = 180^\circ - (52^\circ + 68^\circ) = 60^\circ\ &\theta = \angle BCD = 60^\circ \end{aligned} $$ ## Solve Question 3 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle CAB = \angle CBA = 41^\circ\ &\angle ACB = 180^\circ - (41^\circ + 41^\circ) = 98^\circ\ &\angle ECD = \angle ACB = 98^\circ\ &\angle CED = \angle CDE = \frac{180^\circ - 98^\circ}{2} = 41^\circ\ &\theta = 180^\circ - 41^\circ = 139^\circ \end{aligned} $$ ## Solve Question 4 Using the *Parallel Line Angles*, *Angle Relationships*, and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle BDC = 180^\circ - 116^\circ = 64^\circ\ &\angle DBC = \angle BDC = 64^\circ\ &\angle BCD = 180^\circ - (64^\circ + 64^\circ) = 52^\circ\ &\theta = 180^\circ - 52^\circ = 128^\circ \end{aligned} $$ ## Solve Question 5 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle BCA = 47^\circ - 18^\circ = 29^\circ\ &\angle BAC = 180^\circ - (18^\circ + 29^\circ) = 133^\circ\ &\theta = \angle BAC = 133^\circ \end{aligned} $$ ## Solve Question 6 Using the *Parallel Line Angles*, *Angle Relationships*, and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle ABC = 180^\circ - 77^\circ = 103^\circ\ &\angle BAC = 36^\circ\ &\angle ACB = 180^\circ - (103^\circ + 36^\circ) = 41^\circ\ &\theta = \angle ECD = \angle ACB = 41^\circ \end{aligned} $$ ## Solve Question 7 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle CDE = 180^\circ - 120^\circ = 60^\circ\ &\angle CED = 180^\circ - (60^\circ + 60^\circ) = 60^\circ\ &\theta = \angle CED = 60^\circ \end{aligned} $$ ## Solve Question 8 Using the *Parallel Line Angles*, *Angle Relationships*, and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle ABC = 180^\circ - 135^\circ = 45^\circ\ &\angle ACB = 90^\circ\ &\angle BAC = 180^\circ - (90^\circ + 45^\circ) = 45^\circ\ &\theta = \angle BAC = 45^\circ \end{aligned} $$ ## Solve Question 9 Using the *Parallel Line Angles* and *Triangle Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle BAC = 180^\circ - (45^\circ + 45^\circ) = 90^\circ\ &\theta = 180^\circ - (65^\circ + 90^\circ) = 25^\circ \end{aligned} $$ | No. | Answer | |---|---| | 1 | $\theta = 46^\circ$ | | 2 | $\theta = 60^\circ$ | | 3 | $\theta = 139^\circ$ | | 4 | $\theta = 128^\circ$ | | 5 | $\theta = 133^\circ$ | | 6 | $\theta = 41^\circ$ | | 7 | $\theta = 60^\circ$ | | 8 | $\theta = 45^\circ$ | | 9 | $\theta = 25^\circ$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Angle Relationships" ] }

No.	Answer
2	\(\theta = 60^\circ\)
3	\(\theta = 139^\circ\)
4	\(\theta = 128^\circ\)
5	\(\theta = 133^\circ\)
6	\(\theta = 41^\circ\)
7	\(\theta = 60^\circ\)
8	\(\theta = 45^\circ\)
9	\(\theta = 25^\circ\)

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