Year 7 Maths • AC9M7M05 • Homework
Triangle and Polygon Interior Angles
Show working and give reasons. Use the triangle angle sum of 180° and polygon decomposition rather than guessing from the diagram.
Part A • 10 marks
Short-response questions
State the interior angle sum of a triangle and describe one concrete way to demonstrate it. (2 marks)
A triangle has angles 48°, 67° and x°. Find x. (2 marks)
Write the formula for the interior angle sum of an n-sided polygon. (2 marks)
Calculate the interior angle sum of a hexagon. (2 marks)
A quadrilateral has angles 95°, 80°, 110° and x°. Find x. (2 marks)
Part B • 25 marks
Extended-response questions
A triangle has angles 2x°, 3x° and 4x°. Find x and all three angle sizes. Show working. (5 marks)
Explain why an n-sided polygon can be decomposed into n−2 triangles from one vertex. Use this reasoning to derive 180(n−2). (5 marks)
A pentagon has angles 100°, 120°, 95°, 85° and x°. Find x and explain how you checked your answer. (5 marks)
A polygon has an interior angle sum of 1260°. Determine the number of sides and justify each algebraic step. (5 marks)
A regular octagon has equal interior angles. Find each interior angle and explain why dividing the angle sum by 8 is valid. (5 marks)
Part C • Enrichment
Design and justify
11. A designer wants a convex polygon whose interior angle sum is 2160°. Determine the number of sides. If the polygon is regular, calculate each interior angle. Then explain how triangulation provides a reason for both calculations rather than treating the formula as a rule to memorise.
Teacher / checking section
Answers
1
180°. A valid demonstration is tearing off the three corners of a paper triangle and arranging them along a straight line.
2
x = 180 − 48 − 67 = 65°.
3
180(n−2), equivalent to 180n−360.
4
180(6−2) = 720°.
5
x = 360 − (95+80+110) = 75°.
6
9x=180, so x=20°. The angles are 40°, 60°, 80°.
7
Drawing diagonals from one vertex partitions an n-gon into n−2 non-overlapping triangles. Each contributes 180°, so the sum is 180(n−2).
8
Pentagon sum = 540°. Known angles total 400°, so x=140°. Check: 100+120+95+85+140=540.
9
180(n−2)=1260 → n−2=7 → n=9.
10
Octagon sum = 1080°. Because a regular octagon has 8 equal interior angles, each is 1080÷8=135°.
11
180(n−2)=2160 → n−2=12 → n=14. A regular 14-gon has each interior angle 2160÷14 ≈ 154.3°. Triangulation explains the total because 14−2=12 triangles contribute 12×180°.