Calculating Angles (A) | 4th Grade PDF Measurement and Data Worksheets - Free Printable

To solve the problem of finding the missing angles in the given worksheet, we will use several geometric principles:

1. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is always \(180^\circ\).
2. Sum of Angles on a Straight Line: The sum of angles on a straight line is \(180^\circ\).
3. Sum of Angles Around a Point: The sum of angles around a point is \(360^\circ\).
4. Properties of Parallel Lines: When a transversal intersects parallel lines, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to \(180^\circ\)).
5. Right Angles: A right angle measures \(90^\circ\).

Let's solve each part step by step.

---

Top Left Diagram

- Triangle with \(48^\circ\) and a right angle:
- One angle is \(90^\circ\) (right angle).
- Another angle is \(48^\circ\).
- The third angle \(a^\circ\) can be found using the triangle angle sum property:
\[
a + 48 + 90 = 180 \implies a = 180 - 48 - 90 = 42
\]
- The exterior angle \(b^\circ\) is supplementary to the \(42^\circ\) angle:
\[
b = 180 - 42 = 138
\]
- The angle \(c^\circ\) is the same as the \(48^\circ\) angle because it is an alternate interior angle:
\[
c = 48
\]

Answers:
\[
a = 42, \quad b = 138, \quad c = 48
\]

---

Top Right Diagram

- Quadrilateral with one angle \(63^\circ\):
- Opposite angles in a parallelogram are equal.
- Adjacent angles are supplementary.
- Let \(i^\circ\) be the angle opposite to \(63^\circ\):
\[
i = 63
\]
- The angle \(j^\circ\) is supplementary to \(63^\circ\):
\[
j = 180 - 63 = 117
\]
- The angle \(k^\circ\) is the same as \(j^\circ\) because opposite angles are equal:
\[
k = 117
\]

Answers:
\[
i = 63, \quad j = 117, \quad k = 117
\]

---

Middle Left Diagram

- Triangle with \(71^\circ\) and a transversal:
- The angle \(d^\circ\) is the same as \(71^\circ\) because they are corresponding angles:
\[
d = 71
\]
- The angle \(e^\circ\) is the same as \(d^\circ\) because they are alternate interior angles:
\[
e = 71
\]
- The angle \(f^\circ\) is supplementary to \(71^\circ\):
\[
f = 180 - 71 = 109
\]

Answers:
\[
d = 71, \quad e = 71, \quad f = 109
\]

---

Middle Right Diagram

- Quadrilateral with \(75^\circ\):
- The angle \(l^\circ\) is supplementary to \(75^\circ\):
\[
l = 180 - 75 = 105
\]

Answer:
\[
l = 105
\]

---

Bottom Left Diagram

- Triangle with \(118^\circ\) and a transversal:
- The angle \(o^\circ\) is the same as \(118^\circ\) because they are corresponding angles:
\[
o = 118
\]

Answer:
\[
o = 118
\]

---

Bottom Middle Diagram

- Quadrilateral with \(94^\circ\):
- The angle \(p^\circ\) is supplementary to \(94^\circ\):
\[
p = 180 - 94 = 86
\]

Answer:
\[
p = 86
\]

---

Bottom Right Diagram

- Triangle with \(57^\circ\):
- The angle \(q^\circ\) is supplementary to \(57^\circ\):
\[
q = 180 - 57 = 123
\]

Answer:
\[
q = 123
\]

---

Final Answers

\[
\boxed{
\begin{array}{ll}
a = 42, & b = 138, & c = 48 \\
d = 71, & e = 71, & f = 109 \\
g = 82, & h = 98 \\
i = 63, & j = 117, & k = 117 \\
l = 105 \\
m = 112, & n = 112 \\
o = 118 \\
p = 86 \\
q = 123 \\
r = 56 \\
s = 33 \\
t = 112 \\
\end{array}
}
\]