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Complex Polygon Angles

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Complex Polygon Angles

Talia and Oliver are planning a new fence layout on their dairy farm. Help them calculate missing angles and check their polygon designs. Show your working for calculation questions. Remember: the interior angle sum of an n-sided polygon is (n − 2) × 180°.

Spot the angle strategy

Use what you know about triangles and polygons to choose the best reasoning.

1.Which method can help you work out the interior angle sum of a polygon?

Divide it into triangles from one vertex
Measure only its longest side
Add the lengths of its sides
Divide 360° by the number of sides

2.A quadrilateral has interior angles of 82°, 96°, 101° and x°. Which equation will find x?

x = 360° − (82° + 96° + 101°)
x = 180° − (82° + 96° + 101°)
x = 360° + 82° + 96° + 101°
x = (82° + 96° + 101°) ÷ 3

3.True or false: A hexagon can be divided into four triangles by drawing diagonals from one vertex.

TrueFalse

Calculate the missing angles

The farm’s new fence panels have different polygon shapes. Work carefully and include the angle sum you use.

4.A triangular gate has two interior angles of 48° and 67°. Find its third interior angle.

5.A four-sided holding pen has angles of 85°, 95° and 110°. Calculate the fourth angle.

6.A pentagonal feed-storage area is split into triangles from one vertex. How many triangles are formed, and what is the total of its interior angles?

7.A six-sided silage pad has interior angles of 120°, 115°, 130°, 125° and 110°. Find the sixth angle.

Solve and explain

Use polygon rules to check the farm’s final design.

8.Oliver says an octagon’s interior angles add to 1,080°. Is he correct? Show how you know.

9.Charlotte sketches a polygon and draws diagonals from one vertex to make 7 triangles. How many sides does her polygon have? Explain the rule you use.

10.A five-sided fence panel has angles 100°, 105°, 95° and 120°. Find the fifth angle, then check whether your answer is reasonable.

11.Draw any concave quadrilateral (one with an inward-pointing corner). Add diagonals or another construction to show how you can check that its interior angles still total 360°.