Angles of Triangles Worksheets - Free Printable

The task involves finding the interior angle \( X \) of each triangle using the properties of triangles and basic arithmetic. The key property used here is that the sum of the interior angles of a triangle is always \( 180^\circ \). Let's go through each problem step by step.

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Problem 1:

Given: One angle is \( 36^\circ \), and the triangle is isosceles.
Solution:
- In an isosceles triangle, the two base angles are equal.
- Let the two base angles be \( X \).
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + X + 36^\circ = 180^\circ
\]
\[
2X + 36^\circ = 180^\circ
\]
\[
2X = 180^\circ - 36^\circ
\]
\[
2X = 144^\circ
\]
\[
X = \frac{144^\circ}{2} = 72^\circ
\]
Answer: \( X = 72^\circ \)

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Problem 2:

Given: One angle is \( 54^\circ \), and the triangle is isosceles.
Solution:
- Similar to Problem 1, the two base angles are equal.
- Let the two base angles be \( X \).
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + X + 54^\circ = 180^\circ
\]
\[
2X + 54^\circ = 180^\circ
\]
\[
2X = 180^\circ - 54^\circ
\]
\[
2X = 126^\circ
\]
\[
X = \frac{126^\circ}{2} = 63^\circ
\]
Answer: \( X = 63^\circ \)

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Problem 3:

Given: Two angles are \( 90^\circ \) and \( 63^\circ \).
Solution:
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + 90^\circ + 63^\circ = 180^\circ
\]
\[
X + 153^\circ = 180^\circ
\]
\[
X = 180^\circ - 153^\circ
\]
\[
X = 27^\circ
\]
Answer: \( X = 27^\circ \)

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Problem 4:

Given: Two angles are \( 62^\circ \) and \( 62^\circ \).
Solution:
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + 62^\circ + 62^\circ = 180^\circ
\]
\[
X + 124^\circ = 180^\circ
\]
\[
X = 180^\circ - 124^\circ
\]
\[
X = 56^\circ
\]
Answer: \( X = 56^\circ \)

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Problem 5:

Given: Two angles are \( 36^\circ \) and \( 36^\circ \).
Solution:
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + 36^\circ + 36^\circ = 180^\circ
\]
\[
X + 72^\circ = 180^\circ
\]
\[
X = 180^\circ - 72^\circ
\]
\[
X = 108^\circ
\]
Answer: \( X = 108^\circ \)

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Problem 6:

Given: Two angles are \( 90^\circ \) and \( 45^\circ \).
Solution:
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + 90^\circ + 45^\circ = 180^\circ
\]
\[
X + 135^\circ = 180^\circ
\]
\[
X = 180^\circ - 135^\circ
\]
\[
X = 45^\circ
\]
Answer: \( X = 45^\circ \)

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Problem 7:

Given: Two angles are \( 90^\circ \) and \( 47^\circ \).
Solution:
- The sum of the angles in a triangle is \( 180^\circ \):
\[
X + 90^\circ + 47^\circ = 180^\circ
\]
\[
X + 137^\circ = 180^\circ
\]
\[
X = 180^\circ - 137^\circ
\]
\[
X = 43^\circ
\]
Answer: \( X = 43^\circ \)

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Problem 8:

Given: All three angles are equal (equilateral triangle).
Solution:
- In an equilateral triangle, all angles are equal.
- Each angle is:
\[
X = \frac{180^\circ}{3} = 60^\circ
\]
Answer: \( X = 60^\circ \)

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Final Answer:

\[
\boxed{72^\circ, 63^\circ, 27^\circ, 56^\circ, 108^\circ, 45^\circ, 43^\circ, 60^\circ}
\]