KS4. Geometry. Angles & Symmetry. Alternate & Corresponding Angles ... - Free Printable

To solve the problem of finding the angles marked with letters, we will use properties of angles formed by parallel lines and transversals, as well as properties of triangles. Let's go through each part step by step.

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1.

- Given: \(72^\circ\)
- Angle \(a\) is an alternate interior angle to the given \(72^\circ\) angle.
- Solution: \(a = 72^\circ\)

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2.

- Given: \(82^\circ\)
- Angle \(b\) is a corresponding angle to the given \(82^\circ\) angle.
- Solution: \(b = 82^\circ\)

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3.

- Given: \(100^\circ\)
- Angle \(t\) is a supplementary angle to the given \(100^\circ\) angle (since they form a linear pair).
- Solution: \(t = 180^\circ - 100^\circ = 80^\circ\)

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4.

- Given: \(74^\circ\)
- Angle \(c\) is a vertical angle to the given \(74^\circ\) angle.
- Solution: \(c = 74^\circ\)

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5.

- Given: \(86^\circ\)
- Angle \(y\) is an alternate exterior angle to the given \(86^\circ\) angle.
- Solution: \(y = 86^\circ\)

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6.

- Given: \(92^\circ\)
- Angle \(x\) is a corresponding angle to the given \(92^\circ\) angle.
- Solution: \(x = 92^\circ\)

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7.

- Given: \(95^\circ\) and \(50^\circ\)
- The angles \(x\) and \(y\) are alternate interior angles to the given angles.
- Solution: \(x = 95^\circ\) and \(y = 50^\circ\)

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8.

- Given: \(74^\circ\) and \(93^\circ\)
- Angle \(a\) is an alternate interior angle to the given \(74^\circ\) angle.
- Angle \(b\) is a corresponding angle to the given \(93^\circ\) angle.
- Solution: \(a = 74^\circ\) and \(b = 93^\circ\)

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9.

- Given: \(115^\circ\) and \(77^\circ\)
- Angle \(a\) is a corresponding angle to the given \(115^\circ\) angle.
- Angle \(c\) is an alternate interior angle to the given \(77^\circ\) angle.
- Solution: \(a = 115^\circ\) and \(c = 77^\circ\)

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10.

- Given: \(68^\circ\) and \(42^\circ\)
- The triangle has angles \(68^\circ\), \(42^\circ\), and the third angle (let's call it \(b\)).
- The sum of the angles in a triangle is \(180^\circ\):
\[
68^\circ + 42^\circ + b = 180^\circ
\]
\[
b = 180^\circ - 110^\circ = 70^\circ
\]
- Angle \(a\) is an exterior angle to the triangle, so it is equal to the sum of the two non-adjacent interior angles:
\[
a = 68^\circ + 42^\circ = 110^\circ
\]
- Solution: \(a = 110^\circ\) and \(b = 70^\circ\)

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11.

- Given: \(65^\circ\) and \(130^\circ\)
- The triangle has angles \(65^\circ\), \(z\), and the third angle (let's call it \(y\)).
- The sum of the angles in a triangle is \(180^\circ\):
\[
65^\circ + z + y = 180^\circ
\]
- The angle \(z\) is an exterior angle to the triangle, so it is equal to the sum of the two non-adjacent interior angles:
\[
z = 65^\circ + y
\]
- Since \(z\) is also given as \(130^\circ\):
\[
130^\circ = 65^\circ + y
\]
\[
y = 130^\circ - 65^\circ = 65^\circ
\]
- Substitute \(y = 65^\circ\) back into the triangle angle sum equation:
\[
65^\circ + z + 65^\circ = 180^\circ
\]
\[
z = 180^\circ - 130^\circ = 50^\circ
\]
- Solution: \(y = 65^\circ\) and \(z = 50^\circ\)

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12.

- Given: \(125^\circ\) and \(50^\circ\)
- Angle \(a\) is a corresponding angle to the given \(50^\circ\) angle.
- Angle \(b\) is an alternate interior angle to the given \(125^\circ\) angle.
- Solution: \(a = 50^\circ\) and \(b = 125^\circ\)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & a = 72^\circ \\
2. & b = 82^\circ \\
3. & t = 80^\circ \\
4. & c = 74^\circ \\
5. & y = 86^\circ \\
6. & x = 92^\circ \\
7. & x = 95^\circ, y = 50^\circ \\
8. & a = 74^\circ, b = 93^\circ \\
9. & a = 115^\circ, c = 77^\circ \\
10. & a = 110^\circ, b = 70^\circ \\
11. & y = 65^\circ, z = 50^\circ \\
12. & a = 50^\circ, b = 125^\circ \\
\end{array}
}
\]