Angles in a Triangle Worksheets and Solutions - Free Printable

Let's solve each part of the problem step by step, using geometry principles such as triangle angle sum, properties of parallel lines, isosceles triangles, and quadrilaterals.

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a) Right triangle with angles 90°, 24°, and unknown $ x $

We are given:
- One angle is $ 90^\circ $ (right angle)
- Another angle is $ 24^\circ $
- We need to find $ x $

Sum of angles in a triangle = 180°

$$
x + 90^\circ + 24^\circ = 180^\circ
$$
$$
x = 180^\circ - 90^\circ - 24^\circ = 66^\circ
$$

✔ Answer: $ x = 66^\circ $

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b) Triangle with exterior angle 51°, interior angle 19°, find $ x $

We have:
- An exterior angle of $ 51^\circ $
- Adjacent interior angle is $ 19^\circ $
- The third angle is $ x $

Key idea: The exterior angle equals the sum of the two non-adjacent interior angles.

But here, we can also use triangle angle sum.

Let’s label:
- The triangle has angles: $ x $, $ 19^\circ $, and the third angle adjacent to the $ 51^\circ $ exterior angle.

The exterior angle $ 51^\circ $ is supplementary to its adjacent interior angle:

$$
\text{Adjacent interior angle} = 180^\circ - 51^\circ = 129^\circ
$$

Wait — that can’t be right because then the triangle would have angles $ x $, $ 19^\circ $, and $ 129^\circ $. Sum would be $ x + 19 + 129 = 148 + x $, which must be 180 → $ x = 32^\circ $. But let’s check if this makes sense.

Actually, perhaps the $ 51^\circ $ is an exterior angle, and it's formed by extending one side. Then:

> Exterior angle = sum of two opposite interior angles.

So:

$$
51^\circ = x + 19^\circ
$$

$$
x = 51^\circ - 19^\circ = 32^\circ
$$

✔ Answer: $ x = 32^\circ $

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c) Isosceles triangle with vertex angle $ 50^\circ $, find base angle $ x $

We are told:
- Two sides are marked equal (so it's isosceles)
- Vertex angle = $ 50^\circ $
- Find base angle $ x $

In an isosceles triangle, base angles are equal.

Sum of angles = $ 180^\circ $

$$
x + x + 50^\circ = 180^\circ
$$
$$
2x = 130^\circ
$$
$$
x = 65^\circ
$$

✔ Answer: $ x = 65^\circ $

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d) Quadrilateral-like figure with two angles labeled $ 122^\circ $, and one angle $ x $, with a straight line

We see:
- Two angles of $ 122^\circ $, one on each side of a transversal
- A triangle or polygon with a point where $ x $ is located
- There's a straight line, so angles on a straight line sum to $ 180^\circ $

Looking at the diagram:
- The two $ 122^\circ $ angles are supplementary to the interior angles at the same vertices.
- So the interior angles adjacent to $ 122^\circ $ are:

$$
180^\circ - 122^\circ = 58^\circ
$$

So now, the triangle has two angles of $ 58^\circ $ each?

Wait — actually, the shape looks like a triangle with two exterior angles of $ 122^\circ $, meaning the interior angles are $ 58^\circ $.

But there are two $ 122^\circ $ angles shown, likely both exterior angles.

So if two exterior angles are $ 122^\circ $, then their corresponding interior angles are:

$$
180^\circ - 122^\circ = 58^\circ
$$

Now, the triangle has two interior angles of $ 58^\circ $? But then the third angle $ x $ would be:

$$
x = 180^\circ - 58^\circ - 58^\circ = 64^\circ
$$

Wait — but is that consistent? Let’s think again.

Alternatively, maybe the figure shows a triangle with two external angles of $ 122^\circ $, and we’re to find the internal angle $ x $.

But only one angle $ x $ is marked inside.

Wait — actually, looking carefully: the figure shows a triangle with two straight lines extended, forming two $ 122^\circ $ angles outside.

So:

- At one vertex, exterior angle = $ 122^\circ $ → interior angle = $ 180^\circ - 122^\circ = 58^\circ $
- At another vertex, same thing → interior angle = $ 58^\circ $
- Then the third angle $ x $ is:

$$
x = 180^\circ - 58^\circ - 58^\circ = 64^\circ
$$

✔ Answer: $ x = 64^\circ $

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e) Complex quadrilateral with angles and markings

Given:
- One angle is $ 63^\circ $
- One angle is $ 37^\circ $
- There are tick marks indicating equal sides
- Right angles are marked
- Need to find $ x $

Let’s analyze:

From the diagram:
- It appears to be a parallelogram or kite-like shape, but with a diagonal and right angles.
- There are two pairs of equal sides indicated by tick marks.
- Also, right angles are marked.

Let’s suppose:
- The shape is made of two right triangles joined along a diagonal.
- One triangle has angles: $ 63^\circ $, $ 90^\circ $, and missing angle?
- But wait, there's a $ 37^\circ $ angle shown.

Also, note:
- In a right triangle, if one acute angle is $ 37^\circ $, the other is $ 53^\circ $, since $ 90 - 37 = 53 $
- But here we have $ 63^\circ $, not $ 53^\circ $

Wait — perhaps the $ 63^\circ $ and $ 37^\circ $ are related via parallel lines or isosceles triangles.

Let’s look more carefully.

There are:
- Two equal sides marked with one tick mark
- Two other equal sides marked with double ticks
- A right angle at the top-right corner
- Another right angle at bottom-left
- $ 63^\circ $ at bottom-left
- $ 37^\circ $ near the top
- $ x $ at the bottom-right

Wait — perhaps the figure is a quadrilateral divided into two right triangles by a diagonal.

Let’s assume:
- The quadrilateral has two right angles: one at top-right, one at bottom-left
- At bottom-left: angle $ 63^\circ $ is given, but it's adjacent to a right angle? That doesn't make sense unless the total angle is $ 63^\circ $, which is not right.

Wait — no. The $ 63^\circ $ is inside the quadrilateral, and there's a right angle elsewhere.

Let me reinterpret.

Possibility: The shape is a kite or trapezoid, but with a vertical diagonal.

But notice:
- There is a diagonal from top to bottom
- On the left side: two equal sides (tick marks)
- On the right side: two equal sides (double tick marks)
- Top-right corner: right angle
- Bottom-left corner: $ 63^\circ $
- Near top: $ 37^\circ $
- $ x $ at bottom-right

Let’s suppose:
- The top triangle has a right angle and a $ 37^\circ $ angle
- Then the third angle is $ 90^\circ - 37^\circ = 53^\circ $

But the $ 37^\circ $ is marked at the top, near the right side.

Wait — perhaps the $ 37^\circ $ is part of a right triangle.

Let’s consider the top triangle:
- Right angle at top-right
- Angle $ 37^\circ $ at top-left
- Then the third angle (at the apex) is $ 90^\circ - 37^\circ = 53^\circ $

But the $ 63^\circ $ is at the bottom-left.

Now, the bottom triangle:
- Has a right angle at bottom-left? No — the $ 63^\circ $ is at bottom-left, and a right angle is at top-right, but maybe another right angle?

Wait — the bottom-left corner has a $ 63^\circ $ angle, and a right angle is marked elsewhere.

Wait — perhaps the left side is a triangle with $ 63^\circ $ and a right angle?

No — the $ 63^\circ $ is not adjacent to the right angle.

Alternative idea:

Perhaps the entire quadrilateral has:
- One right angle at top-right
- One right angle at bottom-left?
But bottom-left has $ 63^\circ $, which contradicts.

Wait — the $ 63^\circ $ is part of the angle at bottom-left.

Wait — perhaps the bottom-left angle is $ 63^\circ $, and the top-right is $ 90^\circ $, and there's a diagonal.

Now, the key is the equal sides.

Left side: two equal segments → isosceles triangle?

Right side: two equal segments → another isosceles?

But let’s look for angle chasing.

Suppose the vertical diagonal divides the quadrilateral into two triangles.

Top triangle:
- Right angle at top-right
- $ 37^\circ $ at top-left
- So the third angle (at the bottom of the diagonal) is $ 180^\circ - 90^\circ - 37^\circ = 53^\circ $

Bottom triangle:
- At bottom-left: angle is $ 63^\circ $
- At bottom-right: angle $ x $
- At the top of the diagonal: angle is $ 53^\circ $ (from above)

So sum of angles in bottom triangle:

$$
63^\circ + x + 53^\circ = 180^\circ
$$
$$
x = 180^\circ - 63^\circ - 53^\circ = 64^\circ
$$

But wait — is that correct?

Yes! Because the diagonal splits the quadrilateral, and the angle at the diagonal junction is shared.

So:
- Top triangle: angles $ 90^\circ $, $ 37^\circ $, $ 53^\circ $
- Bottom triangle: angles $ 63^\circ $, $ x $, $ 53^\circ $
- So $ x = 180 - 63 - 53 = 64^\circ $

✔ Answer: $ x = 64^\circ $

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✔ Final Answers:

a) $ x = 66^\circ $
b) $ x = 32^\circ $
c) $ x = 65^\circ $
d) $ x = 64^\circ $
e) $ x = 64^\circ $

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