Exterior angles of a triangle | Rajnish Ranjan | Live - Free Printable

Problem Analysis:

The task involves solving for unknown angles in triangles using the properties of exterior angles and the sum of interior angles of a triangle. Let's solve each part step by step.

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Must Know:

#### 1. First Triangle:
- Given angles: \(50^\circ\) and \(70^\circ\).
- The exterior angle \(x\) is formed by extending one side of the triangle.
- Property Used: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

\[
x = 50^\circ + 70^\circ = 120^\circ
\]

Answer: \( x = 120^\circ \)

#### 2. Second Triangle:
- Given angles: \(65^\circ\) and \(45^\circ\).
- The exterior angle \(x\) is formed by extending one side of the triangle.
- Property Used: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

\[
x = 65^\circ + 45^\circ = 110^\circ
\]

Answer: \( x = 110^\circ \)

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Good to Know:

#### 1. First Triangle:
- Given angles: \(62^\circ\) and \(51^\circ\).
- The exterior angle \(x\) is formed by extending one side of the triangle.
- Step 1: Find the third interior angle of the triangle using the fact that the sum of the interior angles of a triangle is \(180^\circ\).

\[
\text{Third angle} = 180^\circ - 62^\circ - 51^\circ = 67^\circ
\]

- Step 2: Use the exterior angle property to find \(x\).

\[
x = 62^\circ + 67^\circ = 129^\circ
\]

- Step 3: The angle \(y\) is the exterior angle at the other vertex, so:

\[
y = 51^\circ + 67^\circ = 118^\circ
\]

Answers: \( x = 129^\circ \), \( y = 118^\circ \)

#### 2. Second Triangle:
- Given angles: \(70^\circ\) and an exterior angle of \(120^\circ\).
- Step 1: Use the exterior angle property to find the third interior angle.

\[
\text{Third interior angle} = 120^\circ - 70^\circ = 50^\circ
\]

- Step 2: Use the sum of interior angles to find \(x\).

\[
x = 180^\circ - 70^\circ - 50^\circ = 60^\circ
\]

- Step 3: The angle \(y\) is the exterior angle at the other vertex, so:

\[
y = 70^\circ + 60^\circ = 130^\circ
\]

Answers: \( x = 60^\circ \), \( y = 130^\circ \)

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Fun to Know:

#### Triangle with Variables:
- Given angles: \(3y^\circ\), \(7y^\circ\), and \(5y^\circ\) as exterior angles.
- Step 1: Recall that an exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore, the interior angles can be expressed in terms of \(y\).

Let the interior angles be \(A\), \(B\), and \(C\). Then:
\[
\text{Exterior angle at } A = 7y^\circ \implies \text{Interior angle at } A = 180^\circ - 7y^\circ
\]
\[
\text{Exterior angle at } B = 5y^\circ \implies \text{Interior angle at } B = 180^\circ - 5y^\circ
\]
\[
\text{Exterior angle at } C = 3y^\circ \implies \text{Interior angle at } C = 180^\circ - 3y^\circ
\]

- Step 2: Use the fact that the sum of the interior angles of a triangle is \(180^\circ\):

\[
(180^\circ - 7y^\circ) + (180^\circ - 5y^\circ) + (180^\circ - 3y^\circ) = 180^\circ
\]

Simplify:

\[
540^\circ - 15y^\circ = 180^\circ
\]

\[
540^\circ - 180^\circ = 15y^\circ
\]

\[
360^\circ = 15y^\circ
\]

\[
y = 24^\circ
\]

- Step 3: Find the interior angles \(A\), \(B\), and \(C\):

\[
\angle A = 180^\circ - 7y^\circ = 180^\circ - 7(24^\circ) = 180^\circ - 168^\circ = 12^\circ
\]
\[
\angle B = 180^\circ - 5y^\circ = 180^\circ - 5(24^\circ) = 180^\circ - 120^\circ = 60^\circ
\]
\[
\angle C = 180^\circ - 3y^\circ = 180^\circ - 3(24^\circ) = 180^\circ - 72^\circ = 108^\circ
\]

- Step 4: Find the exterior angle at \(B\):

\[
\text{Exterior angle at } B = 180^\circ - \angle B = 180^\circ - 60^\circ = 120^\circ
\]

- Step 5: Find \(x\):

The exterior angle \(x\) is given as \(3y^\circ\):

\[
x = 3y = 3(24^\circ) = 72^\circ
\]

Answers: \( x = 72^\circ \), \( y = 24^\circ \), \( \angle A = 12^\circ \), \( \angle B = 60^\circ \), \( \angle C = 108^\circ \), \( \text{exterior } \angle B = 120^\circ \)

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Final Answers:

1. Must Know:
- \( x = 120^\circ \)
- \( x = 110^\circ \)

2. Good to Know:
- \( x = 129^\circ \), \( y = 118^\circ \)
- \( x = 60^\circ \), \( y = 130^\circ \)

3. Fun to Know:
- \( x = 72^\circ \)
- \( y = 24^\circ \)
- \( \angle A = 12^\circ \)
- \( \angle B = 60^\circ \)
- \( \angle C = 108^\circ \)
- \( \text{exterior } \angle B = 120^\circ \)

\boxed{120, 110, 129, 118, 60, 130, 72, 24, 12, 60, 108, 120}