Missing angles of a triangle worksheet - Free Printable

To solve the problem of finding the missing angles in the given 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. Linear Pair: Two angles that form a straight line sum to \(180^\circ\).
3. Vertical Angles: Vertical angles are congruent.
4. Corresponding and Alternate Angles: When two parallel lines are cut by a transversal, corresponding and alternate angles are equal.

Let's solve each part step by step.

---

Part (a)

The triangle has angles \(21^\circ\), \(125^\circ\), and \(x\). We need to find \(x\), \(y\), and \(z\).

#### Step 1: Find \(x\)
The sum of the angles in a triangle is \(180^\circ\):
\[
21^\circ + 125^\circ + x = 180^\circ
\]
\[
146^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 146^\circ = 34^\circ
\]

#### Step 2: Find \(y\) and \(z\)
The angle \(y\) is an exterior angle to the triangle, and it is equal to the sum of the two non-adjacent interior angles:
\[
y = 125^\circ + 21^\circ = 146^\circ
\]

The angle \(z\) is the third interior angle of the triangle, which we already calculated as \(x = 34^\circ\).

So, the values are:
\[
x = 34^\circ, \quad y = 146^\circ, \quad z = 34^\circ
\]

---

Part (b)

This diagram involves intersecting lines and triangles. We need to find \(a\), \(b\), and \(c\).

#### Step 1: Find \(c\)
The angle \(c\) is part of a linear pair with \(78^\circ\):
\[
c + 78^\circ = 180^\circ
\]
\[
c = 180^\circ - 78^\circ = 102^\circ
\]

#### Step 2: Find \(b\)
The angle \(b\) is part of a triangle with angles \(35^\circ\) and \(113^\circ\):
\[
35^\circ + 113^\circ + b = 180^\circ
\]
\[
148^\circ + b = 180^\circ
\]
\[
b = 180^\circ - 148^\circ = 32^\circ
\]

#### Step 3: Find \(a\)
The angle \(a\) is a vertical angle to \(b\), so:
\[
a = b = 32^\circ
\]

So, the values are:
\[
a = 32^\circ, \quad b = 32^\circ, \quad c = 102^\circ
\]

---

Part (c)

This diagram involves intersecting triangles. We need to find \(f\), \(g\), \(m\), \(n\), and \(p\).

#### Step 1: Find \(f\)
The angle \(f\) is part of a triangle with angles \(71^\circ\) and \(34^\circ\):
\[
71^\circ + 34^\circ + f = 180^\circ
\]
\[
105^\circ + f = 180^\circ
\]
\[
f = 180^\circ - 105^\circ = 75^\circ
\]

#### Step 2: Find \(g\)
The angle \(g\) is an exterior angle to the triangle, and it is equal to the sum of the two non-adjacent interior angles:
\[
g = 34^\circ + 75^\circ = 109^\circ
\]

#### Step 3: Find \(m\)
The angle \(m\) is part of a triangle with angles \(55^\circ\) and \(34^\circ\):
\[
55^\circ + 34^\circ + m = 180^\circ
\]
\[
89^\circ + m = 180^\circ
\]
\[
m = 180^\circ - 89^\circ = 91^\circ
\]

#### Step 4: Find \(n\)
The angle \(n\) is the angle formed by the intersection of the two triangles. Since \(n\) is part of a linear pair with \(34^\circ\):
\[
n + 34^\circ = 180^\circ
\]
\[
n = 180^\circ - 34^\circ = 146^\circ
\]

#### Step 5: Find \(p\)
The angle \(p\) is part of a triangle with angles \(55^\circ\) and \(91^\circ\):
\[
55^\circ + 91^\circ + p = 180^\circ
\]
\[
146^\circ + p = 180^\circ
\]
\[
p = 180^\circ - 146^\circ = 34^\circ
\]

So, the values are:
\[
f = 75^\circ, \quad g = 109^\circ, \quad m = 91^\circ, \quad n = 146^\circ, \quad p = 34^\circ
\]

---

Final Answer

\[
\boxed{
\begin{array}{l}
\text{Part (a): } x = 34^\circ, y = 146^\circ, z = 34^\circ \\
\text{Part (b): } a = 32^\circ, b = 32^\circ, c = 102^\circ \\
\text{Part (c): } f = 75^\circ, g = 109^\circ, m = 91^\circ, n = 146^\circ, p = 34^\circ
\end{array}
}
\]