SOLUTION: Angle relationships to find angle measures - Studypool - Free Printable

Problem Analysis and Solution

The task involves solving for missing angles using angle relationships. Let's go through each problem step by step.

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#### Section 1: Using Angle Relationships

##### Problem 1
- Given: Two intersecting lines with one angle marked as \(112^\circ\).
- To Find: The measure of the opposite angle (\(x\)).
- Solution:
- Opposite angles formed by intersecting lines are equal.
- Therefore, \(x = 112^\circ\).

##### Problem 2
- Given: A right triangle with one angle marked as \(68^\circ\) and another angle marked as \(x\).
- To Find: The measure of \(x\).
- Solution:
- In a triangle, the sum of all interior angles is \(180^\circ\).
- One angle is \(90^\circ\) (right angle), and another is \(68^\circ\).
- Therefore, \(x = 180^\circ - 90^\circ - 68^\circ = 22^\circ\).

##### Problem 3
- Given: Two parallel lines cut by a transversal, with one angle marked as \(124^\circ\) and the corresponding angle marked as \(x\).
- To Find: The measure of \(x\).
- Solution:
- Corresponding angles formed by parallel lines and a transversal are equal.
- Therefore, \(x = 124^\circ\).

##### Problem 4
- Given: A straight line with angles \(x\), \(y\), and \(z\) marked.
- To Find: The measures of \(x\), \(y\), and \(z\).
- Solution:
- Angles on a straight line sum to \(180^\circ\).
- Given \(y = 43^\circ\) and one angle is \(43^\circ\):
- \(x + 43^\circ = 180^\circ \implies x = 137^\circ\).
- \(z + 43^\circ = 180^\circ \implies z = 137^\circ\).

##### Problem 5
- Given: A triangle with one angle marked as \(72^\circ\) and two other angles marked as \(x\) and \(y\).
- To Find: The measures of \(x\), \(y\), and \(z\).
- Solution:
- In a triangle, the sum of all interior angles is \(180^\circ\).
- One angle is \(72^\circ\), and another is \(90^\circ\) (right angle).
- \(x + 72^\circ + 90^\circ = 180^\circ \implies x = 18^\circ\).
- \(y = 72^\circ\) (opposite angle).
- \(z = 180^\circ - 72^\circ - 18^\circ = 90^\circ\).

##### Problem 6
- Given: \(\angle 1\) and \(\angle 2\) are vertical angles, and \(\angle 2 = 105^\circ\).
- To Find: The measure of \(\angle 1\).
- Solution:
- Vertical angles are equal.
- Therefore, \(\angle 1 = 105^\circ\).

##### Problem 7
- Given: \(\angle A\) and \(\angle B\) are complementary angles, and \(\angle A = 42^\circ\).
- To Find: The measure of \(\angle B\).
- Solution:
- Complementary angles sum to \(90^\circ\).
- Therefore, \(\angle B = 90^\circ - 42^\circ = 48^\circ\).

##### Problem 8
- Given: \(\angle P\) and \(\angle Q\) are supplementary angles, and \(\angle Q = 64^\circ\).
- To Find: The measure of \(\angle P\).
- Solution:
- Supplementary angles sum to \(180^\circ\).
- Therefore, \(\angle P = 180^\circ - 64^\circ = 116^\circ\).

##### Problem 9
- Given: \(\angle 1\) and \(\angle 2\) form a linear pair, and \(\angle 1 = 113^\circ\).
- To Find: The measure of \(\angle 2\).
- Solution:
- Linear pairs sum to \(180^\circ\).
- Therefore, \(\angle 2 = 180^\circ - 113^\circ = 67^\circ\).

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#### Section 2: Using Algebra

##### Problem 10
- Given: Two intersecting lines with angles \(3x - 17\) and \(2x + 25\).
- To Find: The value of \(x\).
- Solution:
- Opposite angles formed by intersecting lines are equal.
- Therefore, \(3x - 17 = 2x + 25\).
- Solve for \(x\):
\[
3x - 2x = 25 + 17 \implies x = 42
\]

##### Problem 11
- Given: Two angles in a triangle are \(4x + 12\) and \(3x - 11\), and the third angle is \(x + 6\).
- To Find: The value of \(x\).
- Solution:
- The sum of angles in a triangle is \(180^\circ\).
- Therefore, \((4x + 12) + (3x - 11) + (x + 6) = 180\).
- Simplify:
\[
4x + 3x + x + 12 - 11 + 6 = 180 \implies 8x + 7 = 180 \implies 8x = 173 \implies x = \frac{173}{8}
\]

##### Problem 12
- Given: Two intersecting lines with angles \(5x - 5\) and \(3x + 15\).
- To Find: The value of \(x\).
- Solution:
- Opposite angles formed by intersecting lines are equal.
- Therefore, \(5x - 5 = 3x + 15\).
- Solve for \(x\):
\[
5x - 3x = 15 + 5 \implies 2x = 20 \implies x = 10
\]

##### Problem 13
- Given: Two intersecting lines with angles \(4x - 21\) and \(3x + 18\).
- To Find: The value of \(x\).
- Solution:
- Opposite angles formed by intersecting lines are equal.
- Therefore, \(4x - 21 = 3x + 18\).
- Solve for \(x\):
\[
4x - 3x = 18 + 21 \implies x = 39
\]

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Final Answers

1. \(x = 112^\circ\)
2. \(x = 22^\circ\)
3. \(x = 124^\circ\)
4. \(x = 137^\circ\), \(y = 43^\circ\), \(z = 137^\circ\)
5. \(x = 18^\circ\), \(y = 72^\circ\), \(z = 90^\circ\)
6. \(\angle 1 = 105^\circ\)
7. \(\angle B = 48^\circ\)
8. \(\angle P = 116^\circ\)
9. \(\angle 2 = 67^\circ\)

For the algebra problems:
10. \(x = 42\)
11. \(x = \frac{173}{8}\)
12. \(x = 10\)
13. \(x = 39\)

\boxed{x = 42, \frac{173}{8}, 10, 39}