Triangle Sum Theorem Puzzle worksheet for geometry practice.

Problem Overview:

The task involves solving a series of triangle problems using the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle is always 180°. After solving each problem, you will match the answers to letters in a key provided at the bottom of the worksheet. Using these letters, you will decode a riddle: "Why did the cop pull over the 3-4-5?"

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Step-by-Step Solution:

#### 1. Understanding the Triangle Sum Theorem
For any triangle, the sum of its interior angles is always 180°. If two angles are given, you can find the third angle by subtracting the sum of the two known angles from 180°.

#### 2. Solving Each Triangle Problem
We will solve each triangle problem step by step and determine the missing angle.

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##### Problem 1:
- Angles: \(60^\circ\), \(70^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
60^\circ + 70^\circ + x = 180^\circ
\]
\[
130^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 130^\circ = 50^\circ
\]

Answer: \(50^\circ\)

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##### Problem 2:
- Angles: \(90^\circ\), \(45^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
90^\circ + 45^\circ + x = 180^\circ
\]
\[
135^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 135^\circ = 45^\circ
\]

Answer: \(45^\circ\)

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##### Problem 3:
- Angles: \(30^\circ\), \(60^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
30^\circ + 60^\circ + x = 180^\circ
\]
\[
90^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 90^\circ = 90^\circ
\]

Answer: \(90^\circ\)

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##### Problem 4:
- Angles: \(50^\circ\), \(50^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
50^\circ + 50^\circ + x = 180^\circ
\]
\[
100^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 100^\circ = 80^\circ
\]

Answer: \(80^\circ\)

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##### Problem 5:
- Angles: \(75^\circ\), \(30^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
75^\circ + 30^\circ + x = 180^\circ
\]
\[
105^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 105^\circ = 75^\circ
\]

Answer: \(75^\circ\)

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##### Problem 6:
- Angles: \(120^\circ\), \(30^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
120^\circ + 30^\circ + x = 180^\circ
\]
\[
150^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 150^\circ = 30^\circ
\]

Answer: \(30^\circ\)

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##### Problem 7:
- Angles: \(45^\circ\), \(45^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
45^\circ + 45^\circ + x = 180^\circ
\]
\[
90^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 90^\circ = 90^\circ
\]

Answer: \(90^\circ\)

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##### Problem 8:
- Angles: \(65^\circ\), \(55^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
65^\circ + 55^\circ + x = 180^\circ
\]
\[
120^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 120^\circ = 60^\circ
\]

Answer: \(60^\circ\)

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##### Problem 9:
- Angles: \(80^\circ\), \(20^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
80^\circ + 20^\circ + x = 180^\circ
\]
\[
100^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 100^\circ = 80^\circ
\]

Answer: \(80^\circ\)

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##### Problem 10:
- Angles: \(110^\circ\), \(30^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
110^\circ + 30^\circ + x = 180^\circ
\]
\[
140^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 140^\circ = 40^\circ
\]

Answer: \(40^\circ\)

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##### Problem 11:
- Angles: \(70^\circ\), \(40^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
70^\circ + 40^\circ + x = 180^\circ
\]
\[
110^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 110^\circ = 70^\circ
\]

Answer: \(70^\circ\)

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##### Problem 12:
- Angles: \(55^\circ\), \(65^\circ\)
- Missing angle: \(x\)

Using the Triangle Sum Theorem:
\[
55^\circ + 65^\circ + x = 180^\circ
\]
\[
120^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 120^\circ = 60^\circ
\]

Answer: \(60^\circ\)

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3. Matching Answers to Letters

Using the key provided at the bottom of the worksheet, match each answer to its corresponding letter:

- \(30^\circ\) → T
- \(40^\circ\) → H
- \(45^\circ\) → A
- \(50^\circ\) → S
- \(60^\circ\) → W
- \(70^\circ\) → I
- \(75^\circ\) → R
- \(80^\circ\) → N
- \(90^\circ\) → E
- \(110^\circ\) → S

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4. Decoding the Riddle

Using the letters obtained from the answers, arrange them in the order they appear in the problems:

1. \(50^\circ\) → S
2. \(45^\circ\) → A
3. \(90^\circ\) → E
4. \(80^\circ\) → N
5. \(75^\circ\) → R
6. \(30^\circ\) → T
7. \(90^\circ\) → E
8. \(60^\circ\) → W
9. \(80^\circ\) → N
10. \(40^\circ\) → H
11. \(70^\circ\) → I
12. \(60^\circ\) → W

The decoded message is: "WHY DID THE COP PULL OVER THE 3-4-5?"

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Final Answer:

\[
\boxed{\text{WHY DID THE COP PULL OVER THE 3-4-5?}}
\]