Interior and Exterior angles in a Triangle worksheet - Free Printable

Let's solve each of the problems step by step using geometry principles, especially focusing on triangle angle sums and exterior angles.

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🔴 Must Know:

#### Problem 1 (Left):
Triangle with angles:
- One angle = 50°
- Another = 70°
- Exterior angle labeled $ x^\circ $

We are to find $ x $.

✔ Step 1: Find the third interior angle of the triangle.

Sum of angles in a triangle = 180°

So,
$$
\text{Third angle} = 180^\circ - 50^\circ - 70^\circ = 60^\circ
$$

Now, $ x $ is the exterior angle adjacent to this 60° angle. Since exterior angle + interior angle = 180°,

$$
x = 180^\circ - 60^\circ = 120^\circ
$$

Alternatively, use the Exterior Angle Theorem:
> An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

So,
$$
x = 50^\circ + 70^\circ = 120^\circ
$$

✔ Answer: $ \boxed{x = 120^\circ} $

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#### Problem 2 (Right):
Triangle with angles:
- One angle = 65°
- Another = 45°
- Exterior angle at the third vertex labeled $ x^\circ $

Again, use Exterior Angle Theorem:

The exterior angle $ x $ is equal to the sum of the two non-adjacent interior angles:

$$
x = 65^\circ + 45^\circ = 110^\circ
$$

✔ Answer: $ \boxed{x = 110^\circ} $

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🟡 Good to Know:

#### Problem 3 (Left):
Triangle ABC with:
- $ \angle C = 62^\circ $
- $ \angle A = 51^\circ $
- Point D lies on extension of AB beyond B.
- $ \angle CBD = x^\circ $
- $ \angle BCA = y^\circ $ (Wait — actually, it looks like $ y^\circ $ is marked at point B inside the triangle.)

Wait — let's clarify the diagram.

From the image description:
- Triangle ABC: $ \angle C = 62^\circ $, $ \angle A = 51^\circ $
- Line DB extends from B through A (so D-B-A), so BD is extension of AB backward.
- $ x^\circ $ is the angle at B between D and C → that’s an exterior angle at B, $ \angle DBC = x^\circ $
- $ y^\circ $ is the interior angle at B → $ \angle ABC = y^\circ $

Let’s solve:

✔ Step 1: Find $ y^\circ $ = $ \angle ABC $

Sum of angles in triangle = 180°

$$
\angle ABC = 180^\circ - \angle A - \angle C = 180^\circ - 51^\circ - 62^\circ = 67^\circ
$$

So, $ y = 67^\circ $

✔ Step 2: Find $ x^\circ $ = $ \angle DBC $

This is the exterior angle at B, formed by extending side AB to D.

By Exterior Angle Theorem:
$$
x = \angle A + \angle C = 51^\circ + 62^\circ = 113^\circ
$$

Or, since $ x $ and $ y $ are supplementary (they form a straight line):
$$
x = 180^\circ - y = 180^\circ - 67^\circ = 113^\circ
$$

✔ Answers:
- $ x = \boxed{113^\circ} $
- $ y = \boxed{67^\circ} $

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#### Problem 4 (Right):
Triangle ABC with:
- $ \angle C = 70^\circ $
- $ \angle A = x^\circ $
- $ \angle B = x^\circ $ → So triangle is isosceles with $ \angle A = \angle B $
- Also, $ \angle CBD = y^\circ $, where D is on extension of AB beyond B

We need to find $ x $ and $ y $

✔ Step 1: Use triangle angle sum

$$
\angle A + \angle B + \angle C = 180^\circ
\Rightarrow x + x + 70^\circ = 180^\circ
\Rightarrow 2x = 110^\circ
\Rightarrow x = 55^\circ
$$

So, $ \angle A = \angle B = 55^\circ $

✔ Step 2: Find $ y^\circ $ = $ \angle CBD $, the exterior angle at B

Using Exterior Angle Theorem:
$$
y = \angle A + \angle C = 55^\circ + 70^\circ = 125^\circ
$$

Alternatively, since $ \angle ABC = 55^\circ $, then:
$$
y = 180^\circ - 55^\circ = 125^\circ
$$

✔ Answers:
- $ x = \boxed{55^\circ} $
- $ y = \boxed{125^\circ} $

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✔ Final Answers:

#### Must Know:
1. $ x = \boxed{120^\circ} $
2. $ x = \boxed{110^\circ} $

#### Good to Know:
3. $ x = \boxed{113^\circ},\quad y = \boxed{67^\circ} $
4. $ x = \boxed{55^\circ},\quad y = \boxed{125^\circ} $

Let me know if you'd like a visual explanation or diagrams!