Triangle Angle-Sum Theorem worksheet - Free Printable

To solve the problem of finding the measure of the indicated angle in each triangle, we use the fact that the sum of the interior angles of a triangle is always 180°. Let's go through each problem step by step.

---

Problem 1:

Triangle \( \triangle TUV \)
- Given: \( \angle T = 37^\circ \), \( \angle V = 63^\circ \)
- Find: \( m\angle U \)

Using the triangle angle sum property:
\[
\angle T + \angle U + \angle V = 180^\circ
\]
Substitute the given values:
\[
37^\circ + \angle U + 63^\circ = 180^\circ
\]
Combine like terms:
\[
100^\circ + \angle U = 180^\circ
\]
Solve for \( \angle U \):
\[
\angle U = 180^\circ - 100^\circ = 80^\circ
\]

Answer: \( m\angle U = 80^\circ \)

---

Problem 2:

Triangle \( \triangle ABC \)
- Given: \( \angle A = 46^\circ \), \( \angle C = 90^\circ \) (right angle)
- Find: \( m\angle B \)

Using the triangle angle sum property:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
Substitute the given values:
\[
46^\circ + \angle B + 90^\circ = 180^\circ
\]
Combine like terms:
\[
136^\circ + \angle B = 180^\circ
\]
Solve for \( \angle B \):
\[
\angle B = 180^\circ - 136^\circ = 44^\circ
\]

Answer: \( m\angle B = 44^\circ \)

---

Problem 3:

Triangle \( \triangle PQR \)
- Given: \( \angle P = 51^\circ \), \( \angle R = 36^\circ \)
- Find: \( m\angle Q \)

Using the triangle angle sum property:
\[
\angle P + \angle Q + \angle R = 180^\circ
\]
Substitute the given values:
\[
51^\circ + \angle Q + 36^\circ = 180^\circ
\]
Combine like terms:
\[
87^\circ + \angle Q = 180^\circ
\]
Solve for \( \angle Q \):
\[
\angle Q = 180^\circ - 87^\circ = 93^\circ
\]

Answer: \( m\angle Q = 93^\circ \)

---

Problem 4:

Triangle \( \triangle EFG \)
- Given: \( \angle G = 86^\circ \), \( \angle E = 47^\circ \)
- Find: \( m\angle F \)

Using the triangle angle sum property:
\[
\angle E + \angle F + \angle G = 180^\circ
\]
Substitute the given values:
\[
47^\circ + \angle F + 86^\circ = 180^\circ
\]
Combine like terms:
\[
133^\circ + \angle F = 180^\circ
\]
Solve for \( \angle F \):
\[
\angle F = 180^\circ - 133^\circ = 47^\circ
\]

Answer: \( m\angle F = 47^\circ \)

---

Problem 5:

Triangle \( \triangle XYZ \)
- Given: \( \angle Z = 118^\circ \), \( \angle X = 33^\circ \)
- Find: \( m\angle Y \)

Using the triangle angle sum property:
\[
\angle X + \angle Y + \angle Z = 180^\circ
\]
Substitute the given values:
\[
33^\circ + \angle Y + 118^\circ = 180^\circ
\]
Combine like terms:
\[
151^\circ + \angle Y = 180^\circ
\]
Solve for \( \angle Y \):
\[
\angle Y = 180^\circ - 151^\circ = 29^\circ
\]

Answer: \( m\angle Y = 29^\circ \)

---

Problem 6:

Triangle \( \triangle WUV \)
- Given: \( \angle W = 42^\circ \), \( \angle U = 64^\circ \)
- Find: \( m\angle V \)

Using the triangle angle sum property:
\[
\angle W + \angle U + \angle V = 180^\circ
\]
Substitute the given values:
\[
42^\circ + 64^\circ + \angle V = 180^\circ
\]
Combine like terms:
\[
106^\circ + \angle V = 180^\circ
\]
Solve for \( \angle V \):
\[
\angle V = 180^\circ - 106^\circ = 74^\circ
\]

Answer: \( m\angle V = 74^\circ \)

---

Problem 7:

Triangle \( \triangle MLK \)
- Given: \( \angle M = 60^\circ \), \( \angle L = 60^\circ \)
- Find: \( m\angle K \)

Using the triangle angle sum property:
\[
\angle M + \angle L + \angle K = 180^\circ
\]
Substitute the given values:
\[
60^\circ + 60^\circ + \angle K = 180^\circ
\]
Combine like terms:
\[
120^\circ + \angle K = 180^\circ
\]
Solve for \( \angle K \):
\[
\angle K = 180^\circ - 120^\circ = 60^\circ
\]

Answer: \( m\angle K = 60^\circ \)

---

Problem 8:

Triangle \( \triangle STU \)
- Given: \( \angle S = 35^\circ \), \( \angle T = 35^\circ \)
- Find: \( m\angle U \)

Using the triangle angle sum property:
\[
\angle S + \angle T + \angle U = 180^\circ
\]
Substitute the given values:
\[
35^\circ + 35^\circ + \angle U = 180^\circ
\]
Combine like terms:
\[
70^\circ + \angle U = 180^\circ
\]
Solve for \( \angle U \):
\[
\angle U = 180^\circ - 70^\circ = 110^\circ
\]

Answer: \( m\angle U = 110^\circ \)

---

Problem 9:

Triangle \( \triangle CDE \)
- Given: \( \angle E = 53^\circ \), \( \angle D = 77^\circ \)
- Find: \( m\angle C \)

Using the triangle angle sum property:
\[
\angle C + \angle D + \angle E = 180^\circ
\]
Substitute the given values:
\[
\angle C + 77^\circ + 53^\circ = 180^\circ
\]
Combine like terms:
\[
\angle C + 130^\circ = 180^\circ
\]
Solve for \( \angle C \):
\[
\angle C = 180^\circ - 130^\circ = 50^\circ
\]

Answer: \( m\angle C = 50^\circ \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & m\angle U = 80^\circ \\
2) & m\angle B = 44^\circ \\
3) & m\angle Q = 93^\circ \\
4) & m\angle F = 47^\circ \\
5) & m\angle Y = 29^\circ \\
6) & m\angle V = 74^\circ \\
7) & m\angle K = 60^\circ \\
8) & m\angle U = 110^\circ \\
9) & m\angle C = 50^\circ \\
\end{array}
}
\]