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Problem-solving Pack
MathematicsYear 8 · 8.15 Parallel Lines and Polygons
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
Angles in parallel lines with algebra. Two parallel lines are crossed by a transversal. One marked angle is and the co-interior angle on the same side is .
(a) Set up and solve an equation in .
(b) State the size of both angles and verify they are co-interior.
(c) Find the corresponding angle of the angle on the other parallel line.
(a) Set up and solve an equation in .
(b) State the size of both angles and verify they are co-interior.
(c) Find the corresponding angle of the angle on the other parallel line.
Working space
2Problem 2 of 12
Interior-angle polygon problem. A regular polygon has interior angle 162°.
(a) Find the exterior angle.
(b) How many sides?
(c) Find the sum of interior angles.
(a) Find the exterior angle.
(b) How many sides?
(c) Find the sum of interior angles.
Working space
3Problem 3 of 12
Star polygon problem. A regular pentagon has all its diagonals drawn, forming a 5-pointed star.
(a) Find the interior angle of the regular pentagon.
(b) Find the angle at each star tip.
(c) Show that the five tip angles sum to 180°.
(a) Find the interior angle of the regular pentagon.
(b) Find the angle at each star tip.
(c) Show that the five tip angles sum to 180°.
Working space
4Problem 4 of 12
Two-step parallel-line problem. In a diagram with two parallel lines and , and a transversal:
- The acute angle at is .
- The vertically opposite angle is .
(a) Use vertically-opposite-angles property to find .
(b) Find both angles.
(c) Find the obtuse co-interior angle on .
- The acute angle at is .
- The vertically opposite angle is .
(a) Use vertically-opposite-angles property to find .
(b) Find both angles.
(c) Find the obtuse co-interior angle on .
Working space
5Problem 5 of 12
Mixed polygon angle problem. A pentagon and a triangle share an edge.
(a) Find the interior angles of each (regular).
(b) The shared edge: what is the sum of the two interior angles at one endpoint?
(c) Can a regular pentagon and a regular triangle tile a plane around a vertex?
(a) Find the interior angles of each (regular).
(b) The shared edge: what is the sum of the two interior angles at one endpoint?
(c) Can a regular pentagon and a regular triangle tile a plane around a vertex?
Working space
6Problem 6 of 12
Sum-of-angles identity. Prove that the sum of the exterior angles of any convex polygon equals 360°.
Working space
7Problem 7 of 12
Regular hexagon investigation. A regular hexagon has side .
(a) Find the interior angle.
(b) The hexagon can be split into 6 equilateral triangles. Use this to find the area in terms of .
(c) Find the perimeter.
(a) Find the interior angle.
(b) The hexagon can be split into 6 equilateral triangles. Use this to find the area in terms of .
(c) Find the perimeter.
Working space
8Problem 8 of 12
Mixed angles in a complex figure. In a figure: two parallel lines crossed by a transversal create angles labelled at the upper intersection and at the lower. Given :
(a) Find at the upper intersection.
(b) Find at the lower intersection.
(c) Identify which are alternate, corresponding, co-interior, vertically opposite.
(a) Find at the upper intersection.
(b) Find at the lower intersection.
(c) Identify which are alternate, corresponding, co-interior, vertically opposite.
Working space
9Problem 9 of 12
Irregular polygon angles. A pentagon has angles 100°, 110°, 95°, 130°, .
(a) Find .
(b) Classify the polygon (convex or concave).
(c) If , what would that mean?
(a) Find .
(b) Classify the polygon (convex or concave).
(c) If , what would that mean?
Working space
10Problem 10 of 12
Tiling investigation. A regular polygon tiles the plane if its interior angle divides 360° exactly.
(a) Verify that equilateral triangles, squares, and hexagons can tile.
(b) Show that regular pentagons cannot tile.
(c) Suggest a non-regular pentagon that could.
(a) Verify that equilateral triangles, squares, and hexagons can tile.
(b) Show that regular pentagons cannot tile.
(c) Suggest a non-regular pentagon that could.
Working space
11Problem 11 of 12
Parallel-line angle proof. In a diagram, parallel lines and are crossed by transversal . Show that the sum of the two co-interior angles is 180°.
Working space
12Problem 12 of 12
Polygon side count. A regular polygon has interior angle 144°.
(a) Find the number of sides.
(b) Find the sum of all interior angles.
(c) Find each exterior angle.
(d) Show that the interior angle of a polygon with sides where this polygon has sides is bigger.
(a) Find the number of sides.
(b) Find the sum of all interior angles.
(c) Find each exterior angle.
(d) Show that the interior angle of a polygon with sides where this polygon has sides is bigger.
Working space