Quadrilaterals | FIO

Question 6

Find the remaining angles in the following quadrilaterals.

Solution

Understand the Question

Parallelogram Properties:
- Opposite angles are equal.
- Consecutive (adjacent) angles are supplementary, summing to $180^\circ$ .
Rhombus Properties:
- A rhombus is a parallelogram with all four sides equal.
- Diagonals bisect the vertex angles.
- Each diagonal divides the rhombus into two congruent isosceles triangles.

(i) Find the remaining angles in parallelogram $\text{PEAR}$ where $\angle P = 40^\circ$ .

Step 1 · Find Angles of Parallelogram PEAR

Diagram 1

Consecutive angles in a parallelogram are supplementary: $\angle P + \angle E = 180^\circ$

\begin{aligned} 40^\circ + \angle E &= 180^\circ \\ \angle E &= 180^\circ - 40^\circ = 140^\circ \end{aligned}

$\angle P + \angle R = 180^\circ$

\begin{aligned} 40^\circ + \angle R &= 180^\circ \\ \angle R &= 180^\circ - 40^\circ = 140^\circ \end{aligned}

Opposite angles in a parallelogram are equal: $\angle A = \angle P = 40^\circ$

Answer

(i) $\angle E = 140^\circ, \quad \angle A = 40^\circ, \quad \angle R = 140^\circ$

(ii) Find the remaining angles in parallelogram $\text{PQRS}$ where $\angle P = 110^\circ$ .

Step 1 · Find Angles of Parallelogram PQRS

Diagram 2

Consecutive angles in a parallelogram are supplementary: $\angle P + \angle Q = 180^\circ$

\begin{aligned} 110^\circ + \angle Q &= 180^\circ \\ \angle Q &= 180^\circ - 110^\circ = 70^\circ \end{aligned}

$\angle P + \angle S = 180^\circ$

\begin{aligned} 110^\circ + \angle S &= 180^\circ \\ \angle S &= 180^\circ - 110^\circ = 70^\circ \end{aligned}

Opposite angles in a parallelogram are equal: $\angle R = \angle P = 110^\circ$

Answer

(ii) $\angle Q = 70^\circ, \quad \angle R = 110^\circ, \quad \angle S = 70^\circ$

(iii) Find the remaining angles in rhombus $\text{UVWX}$ with diagonal $\text{XV}$ where $\angle \text{XVU} = 30^\circ$ .

Step 1 · Find Angles of Rhombus UVWX

Diagram 3

In a rhombus, diagonals bisect the vertex angles: $\angle \text{XVW} = \angle \text{XVU} = 30^\circ$

Since $\text{UV} = \text{UX}$ , $\triangle \text{UVX}$ is an isosceles triangle: $\angle \text{UXV} = \angle \text{XVU} = 30^\circ$

Since diagonal $\text{XV}$ bisects $\angle \text{X}$ : $\angle \text{WXV} = \angle \text{UXV} = 30^\circ$

Using angle sum property in $\triangle \text{UVX}$ :

\begin{aligned} \angle \text{VUX} + \angle \text{XVU} + \angle \text{UXV} &= 180^\circ \\ \angle \text{VUX} + 30^\circ + 30^\circ &= 180^\circ \\ \angle \text{VUX} + 60^\circ &= 180^\circ \\ \angle \text{VUX} &= 180^\circ - 60^\circ = 120^\circ \end{aligned}

Opposite angles in a rhombus are equal: $\angle \text{VWX} = \angle \text{VUX} = 120^\circ$

Answer

(iii) $\angle \text{XVW} = 30^\circ, \quad \angle \text{UXV} = 30^\circ, \quad \angle \text{WXV} = 30^\circ, \quad \angle \text{VUX} = 120^\circ, \quad \angle \text{VWX} = 120^\circ$

(iv) Find the remaining angles in rhombus $\text{OIAE}$ with diagonal $\text{OE}$ where $\angle \text{AEO} = 20^\circ$ .

Step 1 · Find Angles of Rhombus OIAE

Diagram 4

In a rhombus, diagonals bisect the vertex angles: $\angle \text{OEI} = \angle \text{AEO} = 20^\circ$

Since $\text{AE} = \text{AO}$ , $\triangle \text{AEO}$ is an isosceles triangle: $\angle \text{AOE} = \angle \text{AEO} = 20^\circ$

Since diagonal $\text{OE}$ bisects $\angle \text{O}$ : $\angle \text{IOE} = \angle \text{AOE} = 20^\circ$

Using angle sum property in $\triangle \text{AEO}$ :

\begin{aligned} \angle \text{EAO} + \angle \text{AEO} + \angle \text{AOE} &= 180^\circ \\ \angle \text{EAO} + 20^\circ + 20^\circ &= 180^\circ \\ \angle \text{EAO} + 40^\circ &= 180^\circ \\ \angle \text{EAO} &= 180^\circ - 40^\circ = 140^\circ \end{aligned}

Opposite angles in a rhombus are equal: $\angle \text{OIA} = \angle \text{EAO} = 140^\circ$

Answer

(iv) $\angle \text{OEI} = 20^\circ, \quad \angle \text{AOE} = 20^\circ, \quad \angle \text{IOE} = 20^\circ, \quad \angle \text{EAO} = 140^\circ, \quad \angle \text{OIA} = 140^\circ$

Common Mistakes

Adjacent vs. Opposite Angles: Confusing consecutive angles (which sum to $180^\circ$ ) with opposite angles (which are equal).
Diagonal Angle Bisection: Forgetting that in a rhombus, each diagonal bisects the corner angles into two equal halves.

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