Angles In Quadrilaterals (Year 6) | CGP Plus - Free Printable

To solve the missing angles in the given quadrilaterals and parallelograms, we will use the following key properties:

1. Sum of interior angles in a quadrilateral: The sum of the interior angles of any quadrilateral is always \(360^\circ\).
2. Properties of parallelograms:
- Opposite angles are equal.
- Adjacent angles are supplementary (sum to \(180^\circ\)).

Let's solve each part step by step.

---

Part 1: Angles in Quadrilaterals

#### Quadrilateral 1
- Given angles: \(67^\circ\), \(90^\circ\), \(90^\circ\)
- Missing angle: \(a\)

Using the sum of interior angles:
\[
67^\circ + 90^\circ + 90^\circ + a = 360^\circ
\]
\[
247^\circ + a = 360^\circ
\]
\[
a = 360^\circ - 247^\circ = 113^\circ
\]

#### Quadrilateral 2
- Given angles: \(74^\circ\), \(117^\circ\)
- Missing angle: \(b\)

Using the sum of interior angles:
\[
74^\circ + 117^\circ + 90^\circ + b = 360^\circ
\]
\[
281^\circ + b = 360^\circ
\]
\[
b = 360^\circ - 281^\circ = 79^\circ
\]

#### Quadrilateral 3
- Given angles: \(55^\circ\), \(111^\circ\), \(107^\circ\)
- Missing angle: \(c\)

Using the sum of interior angles:
\[
55^\circ + 111^\circ + 107^\circ + c = 360^\circ
\]
\[
273^\circ + c = 360^\circ
\]
\[
c = 360^\circ - 273^\circ = 87^\circ
\]

#### Quadrilateral 4
- Given angles: \(47^\circ\), \(89^\circ\), \(82^\circ\)
- Missing angle: \(d\)

Using the sum of interior angles:
\[
47^\circ + 89^\circ + 82^\circ + d = 360^\circ
\]
\[
218^\circ + d = 360^\circ
\]
\[
d = 360^\circ - 218^\circ = 142^\circ
\]

#### Quadrilateral 5
- Given angles: \(79^\circ\), \(38^\circ\), \(27^\circ\)
- Missing angle: \(e\)

Using the sum of interior angles:
\[
79^\circ + 38^\circ + 27^\circ + e = 360^\circ
\]
\[
144^\circ + e = 360^\circ
\]
\[
e = 360^\circ - 144^\circ = 216^\circ
\]

#### Quadrilateral 6
- Given angles: \(48^\circ\), \(239^\circ\), \(34^\circ\)
- Missing angle: \(f\)

Using the sum of interior angles:
\[
48^\circ + 239^\circ + 34^\circ + f = 360^\circ
\]
\[
321^\circ + f = 360^\circ
\]
\[
f = 360^\circ - 321^\circ = 39^\circ
\]

---

Part 2: Angles in Parallelograms

#### Parallelogram 1
- Given angles: \(108^\circ\), \(72^\circ\)
- Missing angles: \(a\), \(b\)

In a parallelogram:
- Opposite angles are equal.
- Adjacent angles are supplementary.

- Since \(a\) is opposite \(108^\circ\):
\[
a = 108^\circ
\]

- Since \(b\) is adjacent to \(108^\circ\):
\[
b + 108^\circ = 180^\circ
\]
\[
b = 180^\circ - 108^\circ = 72^\circ
\]

#### Parallelogram 2
- Given angles: \(57^\circ\)
- Missing angles: \(c\), \(d\), \(e\)

In a parallelogram:
- Opposite angles are equal.
- Adjacent angles are supplementary.

- Since \(c\) is opposite \(57^\circ\):
\[
c = 57^\circ
\]

- Since \(d\) is adjacent to \(57^\circ\):
\[
d + 57^\circ = 180^\circ
\]
\[
d = 180^\circ - 57^\circ = 123^\circ
\]

- Since \(e\) is opposite \(d\):
\[
e = 123^\circ
\]

---

Final Answers

#### Quadrilaterals
\[
a = 113^\circ, \quad b = 79^\circ, \quad c = 87^\circ, \quad d = 142^\circ, \quad e = 216^\circ, \quad f = 39^\circ
\]

#### Parallelograms
\[
a = 108^\circ, \quad b = 72^\circ, \quad c = 57^\circ, \quad d = 123^\circ, \quad e = 123^\circ
\]

\[
\boxed{113^\circ, 79^\circ, 87^\circ, 142^\circ, 216^\circ, 39^\circ, 108^\circ, 72^\circ, 57^\circ, 123^\circ, 123^\circ}
\]