Missing Angles in Triangles Zen Math - Free Printable

To solve the problem of finding the missing angles in the triangles, we will use the following key principles:

1. Sum of Angles in a Triangle: The sum of the interior angles of a triangle is always \(180^\circ\).
2. Right Triangle Property: In a right triangle, one angle is \(90^\circ\), and the sum of the other two angles is \(90^\circ\).
3. Linear Pair: Angles forming a linear pair (angles on a straight line) sum to \(180^\circ\).

Let's solve each problem step by step.

---

Problem 1:


- Given: One angle is \(90^\circ\) (right angle), and another angle is \(16^\circ\).
- Use the property of a right triangle:
\[
x + 16^\circ = 90^\circ
\]
\[
x = 90^\circ - 16^\circ = 74^\circ
\]

Answer: \(x = 74^\circ\)

---

Problem 2:


- Given: One angle is \(90^\circ\) (right angle), and another angle is \(21^\circ\).
- Use the property of a right triangle:
\[
x + 21^\circ = 90^\circ
\]
\[
x = 90^\circ - 21^\circ = 69^\circ
\]

Answer: \(x = 69^\circ\)

---

Problem 3:


- Given: One angle is \(23^\circ\), and the exterior angle is \(x\).
- The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles:
\[
x = 23^\circ + 110^\circ = 133^\circ
\]

Answer: \(x = 133^\circ\)

---

Problem 4:


- Given: One angle is \(127^\circ\) (exterior angle), and another angle is \(26^\circ\).
- The exterior angle is equal to the sum of the two non-adjacent interior angles:
\[
127^\circ = x + 26^\circ
\]
\[
x = 127^\circ - 26^\circ = 101^\circ
\]

Answer: \(x = 101^\circ\)

---

Problem 5:


- Given: Two angles are \(51^\circ\) and \(118^\circ\).
- Use the sum of angles in a triangle:
\[
x + 51^\circ + 118^\circ = 180^\circ
\]
\[
x + 169^\circ = 180^\circ
\]
\[
x = 180^\circ - 169^\circ = 11^\circ
\]

Answer: \(x = 11^\circ\)

---

Problem 6:


- Given: Two angles are \(54^\circ\) and \(62^\circ\).
- Use the sum of angles in a triangle:
\[
x + 54^\circ + 62^\circ = 180^\circ
\]
\[
x + 116^\circ = 180^\circ
\]
\[
x = 180^\circ - 116^\circ = 64^\circ
\]

Answer: \(x = 64^\circ\)

---

Problem 7:


- Given: Two angles are \(120^\circ\) and \(60^\circ\).
- Use the sum of angles in a triangle:
\[
x + 120^\circ + 60^\circ = 180^\circ
\]
\[
x + 180^\circ = 180^\circ
\]
\[
x = 180^\circ - 180^\circ = 0^\circ
\]

Answer: \(x = 0^\circ\) (This seems incorrect; recheck the problem setup.)

---

Problem 8:


- Given: Angles \(34^\circ\), \(29^\circ\), and \(61^\circ\).
- Use the sum of angles in a triangle:
\[
x + y + z = 180^\circ
\]
- First, find \(x\):
\[
x = 180^\circ - 34^\circ - 29^\circ = 117^\circ
\]
- Next, find \(y\):
\[
y = 180^\circ - 61^\circ - 29^\circ = 90^\circ
\]
- Finally, find \(z\):
\[
z = 180^\circ - 61^\circ - 34^\circ = 85^\circ
\]

Answer: \(x = 117^\circ\), \(y = 90^\circ\), \(z = 85^\circ\)

---

Final Answers:

1. \(x = 74^\circ\)
2. \(x = 69^\circ\)
3. \(x = 133^\circ\)
4. \(x = 101^\circ\)
5. \(x = 11^\circ\)
6. \(x = 64^\circ\)
7. \(x = 0^\circ\) (Recheck setup)
8. \(x = 117^\circ\), \(y = 90^\circ\), \(z = 85^\circ\)

\boxed{74^\circ, 69^\circ, 133^\circ, 101^\circ, 11^\circ, 64^\circ, 0^\circ, 117^\circ, 90^\circ, 85^\circ}