Missing Angles in Quadrilaterals - Worksheet - ppt download - Free Printable

Problem Overview:

The task involves identifying the shapes and calculating the missing angles in various quadrilaterals. The key property used here is that the sum of the interior angles of a quadrilateral is always 360°.

We will solve each part step by step.

---

Left Column:

#### Shape a:
- Shape: Rectangle
- Given Angles: 90°, 90°, 90°
- Missing Angle: \( x \)

Since all angles in a rectangle are 90°:
\[
x = 90^\circ
\]

#### Shape b:
- Shape: Parallelogram
- Given Angles: 100°, 80°, 100°
- Missing Angle: \( x \)

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°). The missing angle \( x \) is opposite the 80° angle:
\[
x = 80^\circ
\]

#### Shape c:
- Shape: Trapezoid
- Given Angles: 110°, \( x \), \( y \)
- Missing Angles: \( x \) and \( y \)

Using the property that the sum of the interior angles of a quadrilateral is 360°:
\[
110^\circ + x + y + x = 360^\circ
\]
\[
110^\circ + 2x + y = 360^\circ
\]
\[
2x + y = 250^\circ \quad \text{(Equation 1)}
\]

Additionally, since the trapezoid has one pair of parallel sides, the angles on the same side of the legs are supplementary:
\[
110^\circ + y = 180^\circ
\]
\[
y = 70^\circ
\]

Substitute \( y = 70^\circ \) into Equation 1:
\[
2x + 70^\circ = 250^\circ
\]
\[
2x = 180^\circ
\]
\[
x = 90^\circ
\]

Thus:
\[
x = 90^\circ, \quad y = 70^\circ
\]

#### Shape d:
- Shape: Irregular Quadrilateral
- Given Angles: 45°, 105°, 85°
- Missing Angle: \( x \)

Using the property that the sum of the interior angles of a quadrilateral is 360°:
\[
45^\circ + 105^\circ + 85^\circ + x = 360^\circ
\]
\[
235^\circ + x = 360^\circ
\]
\[
x = 125^\circ
\]

#### Shape e:
- Shape: Irregular Quadrilateral
- Given Angles: \( x \), 103°, 112°, 67°
- Missing Angle: \( x \)

Using the property that the sum of the interior angles of a quadrilateral is 360°:
\[
x + 103^\circ + 112^\circ + 67^\circ = 360^\circ
\]
\[
x + 282^\circ = 360^\circ
\]
\[
x = 78^\circ
\]

#### Shape f:
- Shape: Rhombus
- Given Angles: 105°, 30°, \( x \)
- Missing Angle: \( x \)

In a rhombus, opposite angles are equal, and adjacent angles are supplementary. The missing angle \( x \) is opposite the 30° angle:
\[
x = 30^\circ
\]

#### Shape g:
- Shape: Trapezoid
- Given Angles: 103°, \( x \)
- Missing Angle: \( x \)

Using the property that the sum of the interior angles of a quadrilateral is 360°:
\[
103^\circ + x + x + \text{(other two angles)} = 360^\circ
\]

Since the trapezoid has one pair of parallel sides, the angles on the same side of the legs are supplementary. Let the other two angles be \( y \) and \( z \):
\[
103^\circ + y = 180^\circ \quad \Rightarrow \quad y = 77^\circ
\]
\[
x + z = 180^\circ
\]

Using the sum of the interior angles:
\[
103^\circ + x + x + 77^\circ = 360^\circ
\]
\[
180^\circ + 2x = 360^\circ
\]
\[
2x = 180^\circ
\]
\[
x = 90^\circ
\]

#### Shape h:
- Shape: Triangle within a Quadrilateral
- Given Angles: 15°, 80°, \( x \)
- Missing Angle: \( x \)

The triangle inside the quadrilateral has angles summing to 180°:
\[
15^\circ + 80^\circ + x = 180^\circ
\]
\[
95^\circ + x = 180^\circ
\]
\[
x = 85^\circ
\]

#### Shape i:
- Shape: Intersecting Lines
- Given Angles: \( x \), \( y \), \( z \), 135°
- Missing Angles: \( x \), \( y \), \( z \)

Using the properties of intersecting lines and supplementary angles:
\[
x + 135^\circ = 180^\circ \quad \Rightarrow \quad x = 45^\circ
\]
\[
y = 90^\circ \quad \text{(vertical angles)}
\]
\[
z = 135^\circ \quad \text{(supplementary to } x)
\]

---

Right Column:

#### Shape a:
- Shape: Irregular Quadrilateral
- Given Angles: 125°, 95°, \( x \)
- Missing Angle: \( x \)

Using the property that the sum of the interior angles of a quadrilateral is 360°:
\[
125^\circ + 95^\circ + x + \text{(other angle)} = 360^\circ
\]

The other angle is supplementary to 115°:
\[
115^\circ + \text{(other angle)} = 180^\circ \quad \Rightarrow \quad \text{(other angle)} = 65^\circ
\]

Now:
\[
125^\circ + 95^\circ + x + 65^\circ = 360^\circ
\]
\[
285^\circ + x = 360^\circ
\]
\[
x = 75^\circ
\]

#### Shape b:
- Shape: Parallelogram
- Given Angles: \( x \), \( 2x \)
- Missing Angles: \( x \)

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Using the sum of the interior angles:
\[
x + 2x + x + 2x = 360^\circ
\]
\[
6x = 360^\circ
\]
\[
x = 60^\circ
\]

#### Shape c:
- Shape: Isosceles Trapezoid
- Given Angles: 35°, \( x \)
- Missing Angle: \( x \)

In an isosceles trapezoid, the base angles are equal. The other base angle is also \( x \). Using the sum of the interior angles:
\[
35^\circ + x + 35^\circ + x = 360^\circ
\]
\[
70^\circ + 2x = 360^\circ
\]
\[
2x = 290^\circ
\]
\[
x = 145^\circ
\]

#### Shape d:
- Shape: Irregular Quadrilateral
- Given Angles: \( 2x \), \( 3x \), 40°
- Missing Angles: \( x \)

Using the sum of the interior angles:
\[
2x + 3x + 40^\circ + \text{(other angle)} = 360^\circ
\]

The other angle is supplementary to \( 2x \):
\[
2x + \text{(other angle)} = 180^\circ \quad \Rightarrow \quad \text{(other angle)} = 180^\circ - 2x
\]

Now:
\[
2x + 3x + 40^\circ + (180^\circ - 2x) = 360^\circ
\]
\[
3x + 220^\circ = 360^\circ
\]
\[
3x = 140^\circ
\]
\[
x = \frac{140^\circ}{3}
\]

#### Shape e:
- Shape: Parallelogram
- Given Angles: \( 3x + 25^\circ \), \( x + 15^\circ \)
- Missing Angles: \( x \)

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Using the sum of the interior angles:
\[
(3x + 25^\circ) + (x + 15^\circ) + (3x + 25^\circ) + (x + 15^\circ) = 360^\circ
\]
\[
8x + 80^\circ = 360^\circ
\]
\[
8x = 280^\circ
\]
\[
x = 35^\circ
\]

#### Shape f:
- Shape: Rectangle
- Given Angles: \( \frac{8}{6}x \), \( \frac{6}{5}x \)
- Missing Angles: \( x \)

In a rectangle, opposite sides are equal, and all angles are 90°. The given sides must be equal:
\[
\frac{8}{6}x = \frac{6}{5}x
\]

Cross-multiplying:
\[
8 \cdot 5x = 6 \cdot 6x
\]
\[
40x = 36x
\]
\[
4x = 0 \quad \text{(This is not possible; recheck the problem setup.)}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{a: } x = 90^\circ \\
&\text{b: } x = 80^\circ \\
&\text{c: } x = 90^\circ, y = 70^\circ \\
&\text{d: } x = 125^\circ \\
&\text{e: } x = 78^\circ \\
&\text{f: } x = 30^\circ \\
&\text{g: } x = 90^\circ \\
&\text{h: } x = 85^\circ \\
&\text{i: } x = 45^\circ, y = 90^\circ, z = 135^\circ \\
&\text{a: } x = 75^\circ \\
&\text{b: } x = 60^\circ \\
&\text{c: } x = 145^\circ \\
&\text{d: } x = \frac{140^\circ}{3} \\
&\text{e: } x = 35^\circ \\
&\text{f: } \text{Not solvable as given.}
\end{aligned}
}
\]