Interior and Exterior Angles of A Triangle Worksheet | PDF - Free Printable

To solve the problem, we need to use the properties of interior and exterior angles of a triangle. Here are the key concepts:

1. Sum of Interior Angles of a Triangle: The sum of the interior angles of a triangle is always \(180^\circ\).
2. Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

Let's solve each problem step by step.

---

Problem 1

Given:
- \(\angle ECD = ?\)
- \(\angle DCE = 21^\circ\)
- \(\angle CED = 31^\circ\)

#### Solution:
The exterior angle \(\angle ECD\) is equal to the sum of the two non-adjacent interior angles \(\angle DCE\) and \(\angle CED\):
\[
\angle ECD = \angle DCE + \angle CED = 21^\circ + 31^\circ = 52^\circ
\]

Answer: \(\boxed{52^\circ}\)

---

Problem 2

Given:
- \(\angle TQR = ?\)
- \(\angle T = 55^\circ\)
- \(\angle R = 39^\circ\)

#### Solution:
The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle Q = 180^\circ - \angle T - \angle R = 180^\circ - 55^\circ - 39^\circ = 86^\circ
\]
The exterior angle \(\angle TQR\) is equal to the sum of the two non-adjacent interior angles \(\angle T\) and \(\angle R\):
\[
\angle TQR = \angle T + \angle R = 55^\circ + 39^\circ = 94^\circ
\]

Answer: \(\boxed{94^\circ}\)

---

Problem 3

Given:
- \(\angle MNP = ?\)
- \(\angle N = 77^\circ\)
- \(\angle P = 77^\circ\)

#### Solution:
Since \(\angle N = \angle P = 77^\circ\), the triangle is isosceles. The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle M = 180^\circ - \angle N - \angle P = 180^\circ - 77^\circ - 77^\circ = 26^\circ
\]
The exterior angle \(\angle MNP\) is equal to the sum of the two non-adjacent interior angles \(\angle N\) and \(\angle P\):
\[
\angle MNP = \angle N + \angle P = 77^\circ + 77^\circ = 154^\circ
\]

Answer: \(\boxed{154^\circ}\)

---

Problem 4

Given:
- \(\angle MOQ = ?\)
- \(\angle OMQ = 87^\circ\)
- \(\angle OQM = 63^\circ\)

#### Solution:
The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle MOQ = 180^\circ - \angle OMQ - \angle OQM = 180^\circ - 87^\circ - 63^\circ = 30^\circ
\]
The exterior angle \(\angle MOQ\) is equal to the sum of the two non-adjacent interior angles \(\angle OMQ\) and \(\angle OQM\):
\[
\angle MOQ = \angle OMQ + \angle OQM = 87^\circ + 63^\circ = 150^\circ
\]

Answer: \(\boxed{150^\circ}\)

---

Problem 5

Given:
- \(\angle NPM = ?\)
- \(\angle M = 60^\circ\)
- \(\angle P = 60^\circ\)

#### Solution:
Since \(\angle M = \angle P = 60^\circ\), the triangle is equilateral. All angles in an equilateral triangle are \(60^\circ\). Therefore:
\[
\angle N = 60^\circ
\]
The exterior angle \(\angle NPM\) is equal to the sum of the two non-adjacent interior angles \(\angle M\) and \(\angle P\):
\[
\angle NPM = \angle M + \angle P = 60^\circ + 60^\circ = 120^\circ
\]

Answer: \(\boxed{120^\circ}\)

---

Problem 6

Given:
- \(\angle HFX = ?\)
- \(\angle H = 39^\circ\)
- \(\angle W = 26^\circ\)

#### Solution:
The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle F = 180^\circ - \angle H - \angle W = 180^\circ - 39^\circ - 26^\circ = 115^\circ
\]
The exterior angle \(\angle HFX\) is equal to the sum of the two non-adjacent interior angles \(\angle H\) and \(\angle W\):
\[
\angle HFX = \angle H + \angle W = 39^\circ + 26^\circ = 65^\circ
\]

Answer: \(\boxed{65^\circ}\)

---

Problem 7

Given:
- \(\angle RPQ = ?\)
- \(\angle Q = 51^\circ\)
- \(\angle R = 46^\circ\)

#### Solution:
The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle P = 180^\circ - \angle Q - \angle R = 180^\circ - 51^\circ - 46^\circ = 83^\circ
\]
The exterior angle \(\angle RPQ\) is equal to the sum of the two non-adjacent interior angles \(\angle Q\) and \(\angle R\):
\[
\angle RPQ = \angle Q + \angle R = 51^\circ + 46^\circ = 97^\circ
\]

Answer: \(\boxed{97^\circ}\)

---

Problem 8

Given:
- \(\angle XAC = ?\)
- \(\angle B = 47^\circ\)
- \(\angle C = 60^\circ\)

#### Solution:
The sum of the interior angles of a triangle is \(180^\circ\). Therefore:
\[
\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 47^\circ - 60^\circ = 73^\circ
\]
The exterior angle \(\angle XAC\) is equal to the sum of the two non-adjacent interior angles \(\angle B\) and \(\angle C\):
\[
\angle XAC = \angle B + \angle C = 47^\circ + 60^\circ = 107^\circ
\]

Answer: \(\boxed{107^\circ}\)

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Final Answers

\[
\boxed{52^\circ, 94^\circ, 154^\circ, 150^\circ, 120^\circ, 65^\circ, 97^\circ, 107^\circ}
\]