Find the missing angles in triangles worksheet with 12 problems requiring calculation of unknown angles using the triangle angle sum property.

To solve the problem of finding the missing angles in the triangles, we will use the fundamental property of triangles: the sum of the interior angles of a triangle is always 180°. Let's solve each problem step by step.

---

Problem 1

Given angles: \(70^\circ\), \(45^\circ\)
Find the missing angle \(x\).

\[
x + 70^\circ + 45^\circ = 180^\circ
\]
\[
x + 115^\circ = 180^\circ
\]
\[
x = 180^\circ - 115^\circ
\]
\[
x = 65^\circ
\]

Answer for Problem 1: \(x = 65^\circ\)

---

Problem 2

Given angles: \(82^\circ\), \(35^\circ\)
Find the missing angle \(x\).

\[
x + 82^\circ + 35^\circ = 180^\circ
\]
\[
x + 117^\circ = 180^\circ
\]
\[
x = 180^\circ - 117^\circ
\]
\[
x = 63^\circ
\]

Answer for Problem 2: \(x = 63^\circ\)

---

Problem 3

Given angles: \(58^\circ\), \(86^\circ\)
Find the missing angle \(x\).

\[
x + 58^\circ + 86^\circ = 180^\circ
\]
\[
x + 144^\circ = 180^\circ
\]
\[
x = 180^\circ - 144^\circ
\]
\[
x = 36^\circ
\]

Answer for Problem 3: \(x = 36^\circ\)

---

Problem 4

Given angles: \(61^\circ\), \(52^\circ\)
Find the missing angle \(x\).

\[
x + 61^\circ + 52^\circ = 180^\circ
\]
\[
x + 113^\circ = 180^\circ
\]
\[
x = 180^\circ - 113^\circ
\]
\[
x = 67^\circ
\]

Answer for Problem 4: \(x = 67^\circ\)

---

Problem 5

Given angles: \(50^\circ\) and an exterior angle of \(120^\circ\).
Find the missing angles \(x\) and \(y\).

The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Therefore:
\[
y = 120^\circ - 50^\circ = 70^\circ
\]

Now, using the sum of the interior angles:
\[
x + 50^\circ + y = 180^\circ
\]
\[
x + 50^\circ + 70^\circ = 180^\circ
\]
\[
x + 120^\circ = 180^\circ
\]
\[
x = 180^\circ - 120^\circ
\]
\[
x = 60^\circ
\]

Answer for Problem 5: \(x = 60^\circ\), \(y = 70^\circ\)

---

Problem 6

Given angles: \(77^\circ\), \(64^\circ\)
Find the missing angle \(x\).

\[
x + 77^\circ + 64^\circ = 180^\circ
\]
\[
x + 141^\circ = 180^\circ
\]
\[
x = 180^\circ - 141^\circ
\]
\[
x = 39^\circ
\]

Answer for Problem 6: \(x = 39^\circ\)

---

Problem 7

Given angles: \(13^\circ\), \(29^\circ\)
Find the missing angle \(x\).

\[
x + 13^\circ + 29^\circ = 180^\circ
\]
\[
x + 42^\circ = 180^\circ
\]
\[
x = 180^\circ - 42^\circ
\]
\[
x = 138^\circ
\]

Answer for Problem 7: \(x = 138^\circ\)

---

Problem 8

Given angles: \(64^\circ\), \(38^\circ\)
Find the missing angle \(x\).

\[
x + 64^\circ + 38^\circ = 180^\circ
\]
\[
x + 102^\circ = 180^\circ
\]
\[
x = 180^\circ - 102^\circ
\]
\[
x = 78^\circ
\]

Answer for Problem 8: \(x = 78^\circ\)

---

Problem 9

Given angles: \(81^\circ\), \(69^\circ\)
Find the missing angles \(x\) and \(y\).

First, find \(x\):
\[
x + 81^\circ + 69^\circ = 180^\circ
\]
\[
x + 150^\circ = 180^\circ
\]
\[
x = 180^\circ - 150^\circ
\]
\[
x = 30^\circ
\]

Now, find \(y\). Since \(y\) is an exterior angle:
\[
y = 81^\circ + 30^\circ = 111^\circ
\]

Answer for Problem 9: \(x = 30^\circ\), \(y = 111^\circ\)

---

Problem 10

Given angles: \(90^\circ\), \(45^\circ\)
Find the missing angle \(x\).

\[
x + 90^\circ + 45^\circ = 180^\circ
\]
\[
x + 135^\circ = 180^\circ
\]
\[
x = 180^\circ - 135^\circ
\]
\[
x = 45^\circ
\]

Answer for Problem 10: \(x = 45^\circ\)

---

Problem 11

Given angles: \(35^\circ\), \(15^\circ\)
Find the missing angle \(x\).

\[
x + 35^\circ + 15^\circ = 180^\circ
\]
\[
x + 50^\circ = 180^\circ
\]
\[
x = 180^\circ - 50^\circ
\]
\[
x = 130^\circ
\]

Answer for Problem 11: \(x = 130^\circ\)

---

Problem 12

Given angles: \(90^\circ\), \(45^\circ\)
Find the missing angle \(x\).

\[
x + 90^\circ + 45^\circ = 180^\circ
\]
\[
x + 135^\circ = 180^\circ
\]
\[
x = 180^\circ - 135^\circ
\]
\[
x = 45^\circ
\]

Answer for Problem 12: \(x = 45^\circ\)

---

Final Answers

\[
\boxed{
\begin{array}{ll}
1. & x = 65^\circ \\
2. & x = 63^\circ \\
3. & x = 36^\circ \\
4. & x = 67^\circ \\
5. & x = 60^\circ, \, y = 70^\circ \\
6. & x = 39^\circ \\
7. & x = 138^\circ \\
8. & x = 78^\circ \\
9. & x = 30^\circ, \, y = 111^\circ \\
10. & x = 45^\circ \\
11. & x = 130^\circ \\
12. & x = 45^\circ \\
\end{array}
}
\]