Angle Puzzles: Complete with ease | airSlate SignNow - Free Printable

To solve the problem, we need to determine the measures of each labeled angle in the given diagrams using properties of angles, such as:

1. Sum of angles in a triangle: The sum of the interior angles of a triangle is always \(180^\circ\).
2. Vertical angles: Vertical angles are congruent.
3. Supplementary angles: Two angles that form a straight line are supplementary (sum to \(180^\circ\)).
4. Complementary angles: Two angles that form a right angle are complementary (sum to \(90^\circ\)).
5. Corresponding and alternate angles: When parallel lines are cut by a transversal, corresponding angles and alternate interior/exterior angles are congruent.

Let's solve each diagram step by step.

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Diagram 1

#### Given:
- Several angles are labeled with specific measures.
- We need to find the measures of angles \(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q\).

#### Step-by-Step Solution:

1. Angle \(a\):
- \(a\) is part of a triangle where one angle is \(90^\circ\) and another is \(23^\circ\).
- Using the triangle angle sum property: \(a + 90^\circ + 23^\circ = 180^\circ\).
- \(a = 180^\circ - 90^\circ - 23^\circ = 67^\circ\).

2. Angle \(b\):
- \(b\) is vertically opposite to the angle labeled \(122^\circ\).
- Therefore, \(b = 122^\circ\).

3. Angle \(c\):
- \(c\) is part of a triangle where one angle is \(53^\circ\) and another is \(90^\circ\).
- Using the triangle angle sum property: \(c + 53^\circ + 90^\circ = 180^\circ\).
- \(c = 180^\circ - 53^\circ - 90^\circ = 37^\circ\).

4. Angle \(d\):
- \(d\) is vertically opposite to the angle labeled \(37^\circ\).
- Therefore, \(d = 37^\circ\).

5. Angle \(e\):
- \(e\) is part of a triangle where one angle is \(53^\circ\) and another is \(90^\circ\).
- Using the triangle angle sum property: \(e + 53^\circ + 90^\circ = 180^\circ\).
- \(e = 180^\circ - 53^\circ - 90^\circ = 37^\circ\).

6. Angle \(f\):
- \(f\) is vertically opposite to the angle labeled \(37^\circ\).
- Therefore, \(f = 37^\circ\).

7. Angle \(g\):
- \(g\) is part of a triangle where one angle is \(114^\circ\) and another is \(37^\circ\).
- Using the triangle angle sum property: \(g + 114^\circ + 37^\circ = 180^\circ\).
- \(g = 180^\circ - 114^\circ - 37^\circ = 29^\circ\).

8. Angle \(h\):
- \(h\) is vertically opposite to the angle labeled \(29^\circ\).
- Therefore, \(h = 29^\circ\).

9. Angle \(i\):
- \(i\) is part of a triangle where one angle is \(114^\circ\) and another is \(29^\circ\).
- Using the triangle angle sum property: \(i + 114^\circ + 29^\circ = 180^\circ\).
- \(i = 180^\circ - 114^\circ - 29^\circ = 37^\circ\).

10. Angle \(j\):
- \(j\) is vertically opposite to the angle labeled \(37^\circ\).
- Therefore, \(j = 37^\circ\).

11. Angle \(k\):
- \(k\) is part of a triangle where one angle is \(90^\circ\) and another is \(37^\circ\).
- Using the triangle angle sum property: \(k + 90^\circ + 37^\circ = 180^\circ\).
- \(k = 180^\circ - 90^\circ - 37^\circ = 53^\circ\).

12. Angle \(l\):
- \(l\) is vertically opposite to the angle labeled \(53^\circ\).
- Therefore, \(l = 53^\circ\).

13. Angle \(m\):
- \(m\) is part of a triangle where one angle is \(90^\circ\) and another is \(53^\circ\).
- Using the triangle angle sum property: \(m + 90^\circ + 53^\circ = 180^\circ\).
- \(m = 180^\circ - 90^\circ - 53^\circ = 37^\circ\).

14. Angle \(n\):
- \(n\) is vertically opposite to the angle labeled \(37^\circ\).
- Therefore, \(n = 37^\circ\).

15. Angle \(o\):
- \(o\) is part of a triangle where one angle is \(90^\circ\) and another is \(37^\circ\).
- Using the triangle angle sum property: \(o + 90^\circ + 37^\circ = 180^\circ\).
- \(o = 180^\circ - 90^\circ - 37^\circ = 53^\circ\).

16. Angle \(p\):
- \(p\) is vertically opposite to the angle labeled \(53^\circ\).
- Therefore, \(p = 53^\circ\).

17. Angle \(q\):
- \(q\) is part of a triangle where one angle is \(90^\circ\) and another is \(53^\circ\).
- Using the triangle angle sum property: \(q + 90^\circ + 53^\circ = 180^\circ\).
- \(q = 180^\circ - 90^\circ - 53^\circ = 37^\circ\).

#### Final Answers for Diagram 1:
\[
\boxed{
a = 67^\circ, b = 122^\circ, c = 37^\circ, d = 37^\circ, e = 37^\circ, f = 37^\circ, g = 29^\circ, h = 29^\circ, i = 37^\circ, j = 37^\circ, k = 53^\circ, l = 53^\circ, m = 37^\circ, n = 37^\circ, o = 53^\circ, p = 53^\circ, q = 37^\circ
}
\]

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Diagram 2

#### Given:
- Several angles are labeled with specific measures.
- We need to find the measures of angles \(a, b, c, d, e, f, g, h, i, j, k, l, m, n\).

#### Step-by-Step Solution:

1. Angle \(a\):
- \(a\) is part of a triangle where one angle is \(78^\circ\) and another is \(58^\circ\).
- Using the triangle angle sum property: \(a + 78^\circ + 58^\circ = 180^\circ\).
- \(a = 180^\circ - 78^\circ - 58^\circ = 44^\circ\).

2. Angle \(b\):
- \(b\) is vertically opposite to the angle labeled \(44^\circ\).
- Therefore, \(b = 44^\circ\).

3. Angle \(c\):
- \(c\) is part of a triangle where one angle is \(78^\circ\) and another is \(44^\circ\).
- Using the triangle angle sum property: \(c + 78^\circ + 44^\circ = 180^\circ\).
- \(c = 180^\circ - 78^\circ - 44^\circ = 58^\circ\).

4. Angle \(d\):
- \(d\) is vertically opposite to the angle labeled \(58^\circ\).
- Therefore, \(d = 58^\circ\).

5. Angle \(e\):
- \(e\) is part of a triangle where one angle is \(78^\circ\) and another is \(58^\circ\).
- Using the triangle angle sum property: \(e + 78^\circ + 58^\circ = 180^\circ\).
- \(e = 180^\circ - 78^\circ - 58^\circ = 44^\circ\).

6. Angle \(f\):
- \(f\) is vertically opposite to the angle labeled \(44^\circ\).
- Therefore, \(f = 44^\circ\).

7. Angle \(g\):
- \(g\) is part of a triangle where one angle is \(78^\circ\) and another is \(44^\circ\).
- Using the triangle angle sum property: \(g + 78^\circ + 44^\circ = 180^\circ\).
- \(g = 180^\circ - 78^\circ - 44^\circ = 58^\circ\).

8. Angle \(h\):
- \(h\) is vertically opposite to the angle labeled \(58^\circ\).
- Therefore, \(h = 58^\circ\).

9. Angle \(i\):
- \(i\) is part of a triangle where one angle is \(78^\circ\) and another is \(58^\circ\).
- Using the triangle angle sum property: \(i + 78^\circ + 58^\circ = 180^\circ\).
- \(i = 180^\circ - 78^\circ - 58^\circ = 44^\circ\).

10. Angle \(j\):
- \(j\) is vertically opposite to the angle labeled \(44^\circ\).
- Therefore, \(j = 44^\circ\).

11. Angle \(k\):
- \(k\) is part of a triangle where one angle is \(78^\circ\) and another is \(44^\circ\).
- Using the triangle angle sum property: \(k + 78^\circ + 44^\circ = 180^\circ\).
- \(k = 180^\circ - 78^\circ - 44^\circ = 58^\circ\).

12. Angle \(l\):
- \(l\) is vertically opposite to the angle labeled \(58^\circ\).
- Therefore, \(l = 58^\circ\).

13. Angle \(m\):
- \(m\) is part of a triangle where one angle is \(78^\circ\) and another is \(58^\circ\).
- Using the triangle angle sum property: \(m + 78^\circ + 58^\circ = 180^\circ\).
- \(m = 180^\circ - 78^\circ - 58^\circ = 44^\circ\).

14. Angle \(n\):
- \(n\) is vertically opposite to the angle labeled \(44^\circ\).
- Therefore, \(n = 44^\circ\).

#### Final Answers for Diagram 2:
\[
\boxed{
a = 44^\circ, b = 44^\circ, c = 58^\circ, d = 58^\circ, e = 44^\circ, f = 44^\circ, g = 58^\circ, h = 58^\circ, i = 44^\circ, j = 44^\circ, k = 58^\circ, l = 58^\circ, m = 44^\circ, n = 44^\circ
}
\]

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Final Answer:

\[
\boxed{
\text{Diagram 1: } a = 67^\circ, b = 122^\circ, c = 37^\circ, d = 37^\circ, e = 37^\circ, f = 37^\circ, g = 29^\circ, h = 29^\circ, i = 37^\circ, j = 37^\circ, k = 53^\circ, l = 53^\circ, m = 37^\circ, n = 37^\circ, o = 53^\circ, p = 53^\circ, q = 37^\circ \\
\text{Diagram 2: } a = 44^\circ, b = 44^\circ, c = 58^\circ, d = 58^\circ, e = 44^\circ, f = 44^\circ, g = 58^\circ, h = 58^\circ, i = 44^\circ, j = 44^\circ, k = 58^\circ, l = 58^\circ, m = 44^\circ, n = 44^\circ
}
\]