For school sports day, a hexagonal post cap holds a triangular banner. Five of the post cap's interior angles are recorded in the table below. Corner 1 2 3 ...
Question 1 Report
For school sports day, a hexagonal post cap holds a triangular banner. Five of the post cap's interior angles are recorded in the table below.
| Corner | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Interior angle | 125 | 110 | 135 | 118 | 122 | x |
- A hexagon's interior angles total how many degrees? (1)
- Use this total to work out \(x\). (2)
- Calculate the exterior angle at that sixth corner. (1)
- The banner is triangular, with one angle equal to this exterior angle and a second angle of \(55^{\circ}\). Work out the banner's third angle. (2)
- Decide, with a reason, whether the banner is isosceles. (1)
Answer Details
This question combines a polygon's interior angle sum, the exterior/interior angle relationship, and a triangle's angle sum with an isosceles check.
(a) The interior angles of a hexagon sum to \((6-2) \times 180 = 720^\circ\). [1 mark]
(b) Setting the five known angles plus \(x\) equal to this total: \(125+110+135+118+122+x = 720\), so \(610 + x = 720\), giving \(x = 110\). [2 marks]
(c) The exterior angle at that corner and the interior angle \(x\) lie on a straight line, so the exterior angle is \(180 - 110 = 70^\circ\). [1 mark]
(d) Using this \(70^\circ\) as one angle of the banner, with a second angle of \(55^\circ\), the banner's third angle is \(180 - 70 - 55 = 55^\circ\). [2 marks]
(e) Since two of the banner's angles are both \(55^\circ\), yes, the banner is isosceles (equal angles mean the sides opposite them are also equal). [1 mark]
This question links three separate angle facts in sequence: a hexagon's fixed interior-angle total, the straight-line relationship between interior and exterior angles, and the triangle angle sum, each feeding the next calculation.
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