Finding Unknown Angles in Four-Sided Figures

Apply your knowledge of parallelograms, rhombuses, and trapeziums to solve problems

Quick Review: Key Properties

Parallelogram

Opposite angles equal
Adjacent angles add up to 180°

Rhombus

4 equal sides
Same angle properties as parallelogram

Trapezium

1 pair of parallel sides
The two angles on each slanted side add up to 180°

Strategy for Finding Unknown Angles

Identify the shape - Is it a parallelogram, rhombus, or trapezium?
Find parallel sides - Look for arrow markers
Apply the correct property:
- Opposite angles equal (parallelogram/rhombus)
- Adjacent angles sum to 180° (parallelogram/rhombus)
- The two angles on each slanted side, between the parallel sides, sum to 180° (trapezium)
Check your answer - All angles should sum to 360°

Parallelogram Problems

Problem 1: Using opposite angles

EFGH is a parallelogram. ∠HEF = 110°. Find ∠j, ∠k, and ∠m.

Find angles j, k, and m

ShowHide solution

Finding ∠k: ∠k is opposite to ∠HEF

$\angle k = \angle HEF = 110°$

Finding ∠m: ∠m is adjacent to ∠HEF

$\angle m = 180° - 110° = 70°$

Finding ∠j: ∠j is opposite to ∠m

$\angle j = \angle m = 70°$

Answer: ∠j = 70°, ∠k = 110°, ∠m = 70°

Check: 110° + 70° + 110° + 70° = 360° ✓

Problem 2: Using adjacent angles

PQRS is a parallelogram. ∠PQR = 72°. Find ∠a, ∠b, and ∠c.

Find angles a, b, and c

ShowHide solution

Finding ∠a: ∠a is adjacent to ∠PQR

$\angle a = 180° - 72° = 108°$

Finding ∠b: ∠b is opposite to ∠a

$\angle b = \angle a = 108°$

Finding ∠c: ∠c is opposite to ∠PQR

$\angle c = \angle PQR = 72°$

Answer: ∠a = 108°, ∠b = 108°, ∠c = 72°

Check: 108° + 72° + 108° + 72° = 360° ✓

Rhombus Problems

Problem 3: Finding multiple angles

TUVW is a rhombus. ∠UVW = 128°. Find ∠e, ∠f, and ∠g.

Find angles e, f, and g

ShowHide solution

Finding ∠e: Opposite angles are equal

$\angle e = \angle UVW = 128°$

Finding ∠f: Adjacent angles sum to 180°

$\angle f = 180° - 128° = 52°$

Finding ∠g: Opposite to ∠f

$\angle g = \angle f = 52°$

Answer: ∠e = 128°, ∠f = 52°, ∠g = 52°

Check: 128° + 52° + 128° + 52° = 360° ✓

Problem 4: Diamond-shaped rhombus

KLMN is a rhombus. ∠KLM = 124°. Find ∠LMN and ∠MNK.

Find ∠LMN and ∠MNK

ShowHide solution

Step 1: Find ∠LMN (adjacent to ∠KLM)

$\angle LMN = 180° - 124° = 56°$

Step 2: Find ∠MNK (opposite to ∠KLM)

$\angle MNK = \angle KLM = 124°$

Step 3: Check

∠NKL is opposite to ∠LMN, so it is also 56°. The four angles should add up to 360°:

$56° + 124° + 56° + 124° = 360°$ ✓

Answer: $\angle LMN = 56°$ , $\angle MNK = 124°$

Trapezium Problems

Problem 5: Angles on the same slanted side

PQRS is a trapezium. PQ // RS. ∠PQR = 108°. Find ∠QRS.

Find ∠QRS

ShowHide solution

Property: In a trapezium, the two angles on the same slanted side add up to 180°

∠PQR and ∠QRS are on the same slanted side, QR (right side), between the parallel sides

$\angle QRS = 180° - 108°$

Answer: $\angle QRS = 72°$

Problem 6: Multiple unknown angles

WXYZ is a trapezium. WX // ZY. ∠WXY = 118° and ∠WZY = 69°. Find ∠a and ∠c.

Find ∠a and ∠c

ShowHide solution

Finding ∠a: On the same slanted side as ∠WZY (left side)

$\angle a = 180° - 69° = 111°$

Finding ∠c: On the same slanted side as ∠WXY (right side)

$\angle c = 180° - 118° = 62°$

Answer: ∠a = 111°, ∠c = 62°

Check: 69° + 62° + 118° + 111° = 360° ✓

Challenge Problems

Challenge 1: Parallelogram

ABCD is a parallelogram. ∠ABC = 118°. Find ∠m and ∠n.

Find ∠m and ∠n

ShowHide solution

Finding ∠m: ∠m is adjacent to ∠ABC

$\angle m = 180° - 118° = 62°$

Finding ∠n: ∠n is opposite to ∠ABC

$\angle n = \angle ABC = 118°$

Answer: ∠m = 62°, ∠n = 118°

Challenge 2: Finding all trapezium angles

RSTU is a trapezium. RS // UT. ∠URS = 130° and ∠STU = 75°. Find ∠RST and ∠TUR.

Find ∠RST and ∠TUR

ShowHide solution

Finding ∠TUR: On the same slanted side as ∠URS (left side)

$\angle TUR = 180° - 130° = 50°$

Finding ∠RST: On the same slanted side as ∠STU (right side)

$\angle RST = 180° - 75° = 105°$

Answer: ∠TUR = 50°, ∠RST = 105°

Check: 130° + 105° + 75° + 50° = 360° ✓

Summary: Which Property to Use?

Shape	Property	When to Use
Parallelogram	Opposite angles equal	Given one angle, find the angle diagonally across
Parallelogram	Adjacent angles add up to 180°	Given one angle, find the angle next to it
Rhombus	Same as parallelogram	All parallelogram properties apply
Rhombus	4 equal sides	Finding lengths, not angles
Trapezium	The two angles on each slanted side add up to 180°	Given one angle, find the other angle on the same slanted side

Key Takeaways

Always identify the shape first - parallelogram, rhombus, or trapezium
Look for parallel sides (marked with arrows)
For parallelograms and rhombuses: opposite angles are equal, adjacent angles sum to 180°
For trapeziums: the two angles on the same slanted side (between the parallel sides) sum to 180°
Always check your answer - all four angles should sum to 360°
If there are triangles inside the shape, use angle sum = 180° for the triangle too