Question

practice exercises

find the value of x.

1.
find the values of x and y.

4.

7.

2.

5.

8.

3.

6.

9.

Explanation:

<pre_analysis>
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"step_cot",
"explain_then_answer",
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],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Triangle Angle-Sum Theorem",
"Exterior Angle Theorem"
],
"current_concepts": [
"Triangle Angle-Sum Theorem",
"Exterior Angle Theorem",
"Parallel Line Properties"
]
}
</pre_analysis>

<reasoning>

Solve Questions 1 to 3 for the value of x

\[

$$\begin{aligned} &\text{Question 1: } 59^\circ + 33^\circ = 4x \implies 92 = 4x \implies x = 23\\ &\text{Question 2: } 148^\circ = 52^\circ + (180^\circ - 6x) \implies 148 = 232 - 6x \implies 6x = 84 \implies x = 14\\ &\text{Question 3: } 2x + 45 = 3x + 20 \implies x = 25 \end{aligned}$$

\]

Solve Questions 4 to 6 for the values of x and y

\[

$$\begin{aligned} &\text{Question 4: } x = 108 \text{ (corresponding angles), } 3y = 48 + x \implies 3y = 48 + 108 \implies 3y = 156 \implies y = 52\\ &\text{Question 5: } x + 90 + 42 = 180 \implies x = 48, \quad y = x + 90 \implies y = 48 + 90 \implies y = 138\\ &\text{Question 6: } x = 40 + 32 = 72, \quad y = 40 \text{ (alternate interior angles)} \end{aligned}$$

\]

Solve Questions 7 to 9 for the values of x and y

\[

$$\begin{aligned} &\text{Question 7: } x + 118 = 180 \implies x = 62, \quad y + x + 65 = 180 \implies y + 62 + 65 = 180 \implies y = 53\\ &\text{Question 8: } x = 62 \text{ (alternate interior angles), } y + 38 + x = 180 \implies y + 38 + 62 = 180 \implies y = 80\\ &\text{Question 9: } 2x + (5x - 14) + 75 = 180 \implies 7x + 61 = 180 \implies 7x = 119 \implies x = 17\\ &y = 2x + (5x - 14) \implies y = 7x - 14 \implies y = 7(17) - 14 \implies y = 105 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(x = 23\)

Question 2

\(x = 14\)

Question 3

\(x = 25\)

Question 4

\(x = 108\), \(y = 52\)

Question 5

\(x = 48\), \(y = 138\)

Question 6

\(x = 72\), \(y = 40\)

Question 7

\(x = 62\), \(y = 53\)

Question 8

\(x = 62\), \(y = 80\)

Question 9

\(x = 17\), \(y = 105\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Triangle Angle-Sum Theorem"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
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"quick"
],
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"concepts_used": [],
"new_concepts": [
"Triangle Angle-Sum Theorem",
"Exterior Angle Theorem"
],
"current_concepts": [
"Triangle Angle-Sum Theorem",
"Exterior Angle Theorem",
"Parallel Line Properties"
]
}
</pre_analysis>

<reasoning>

Solve Questions 1 to 3 for the value of x

\[

$$\begin{aligned} &\text{Question 1: } 59^\circ + 33^\circ = 4x \implies 92 = 4x \implies x = 23\\ &\text{Question 2: } 148^\circ = 52^\circ + (180^\circ - 6x) \implies 148 = 232 - 6x \implies 6x = 84 \implies x = 14\\ &\text{Question 3: } 2x + 45 = 3x + 20 \implies x = 25 \end{aligned}$$

\]

Solve Questions 4 to 6 for the values of x and y

\[

$$\begin{aligned} &\text{Question 4: } x = 108 \text{ (corresponding angles), } 3y = 48 + x \implies 3y = 48 + 108 \implies 3y = 156 \implies y = 52\\ &\text{Question 5: } x + 90 + 42 = 180 \implies x = 48, \quad y = x + 90 \implies y = 48 + 90 \implies y = 138\\ &\text{Question 6: } x = 40 + 32 = 72, \quad y = 40 \text{ (alternate interior angles)} \end{aligned}$$

\]

Solve Questions 7 to 9 for the values of x and y

\[

$$\begin{aligned} &\text{Question 7: } x + 118 = 180 \implies x = 62, \quad y + x + 65 = 180 \implies y + 62 + 65 = 180 \implies y = 53\\ &\text{Question 8: } x = 62 \text{ (alternate interior angles), } y + 38 + x = 180 \implies y + 38 + 62 = 180 \implies y = 80\\ &\text{Question 9: } 2x + (5x - 14) + 75 = 180 \implies 7x + 61 = 180 \implies 7x = 119 \implies x = 17\\ &y = 2x + (5x - 14) \implies y = 7x - 14 \implies y = 7(17) - 14 \implies y = 105 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(x = 23\)

Question 2

\(x = 14\)

Question 3

\(x = 25\)

Question 4

\(x = 108\), \(y = 52\)

Question 5

\(x = 48\), \(y = 138\)

Question 6

\(x = 72\), \(y = 40\)

Question 7

\(x = 62\), \(y = 53\)

Question 8

\(x = 62\), \(y = 80\)

Question 9

\(x = 17\), \(y = 105\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Triangle Angle-Sum Theorem"
]
}
</post_analysis>