Corrigé
8.15 Parallel Lines and Polygons
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Two lines cross. One of the four angles is 47°. State the other three angles and name the relationship for each. | 47° (vertically opposite), 133°, 133° |
| 2 | A transversal crosses two parallel lines. One alternate (Z) angle is 75°. Find the other. | 75° |
| 3 | A transversal crosses two parallel lines. A corresponding (F) angle is 62°. Find the other. | 62° |
| 4 | Two co-interior (allied / "C") angles between parallel lines are 110° and $x$. Find $x$. | $x = 70°$ |
| 5 | Find the sum of the interior angles of an $n$-sided polygon with $n = 6$. | 720° |
| 6 | Find one interior angle of a regular 6-gon. | 120° |
| 7 | Find the exterior angle of a regular 6-gon. | 60° |
| 8 | In a triangle, two angles are 80° and 60°. Find the third. | 40° |
| 9 | A pentagon has angles 100°, 110°, 95°, 105° and $x$. Find $x$. | 130° |
| 10 | A regular octagon has 8 interior angles. State the size of each. | 135° |
| 11 | Two parallel lines crossed by a transversal: the acute angle on the upper line is 58°. Find the corresponding angle on the lower line. | 58° |
| 12 | Two parallel lines have a transversal. One angle is 70°; another (alternate) is $(x + 15)$°. Find $x$. | $x = 55$ |
| 13 | The exterior angle of a triangle is 110°. The two non-adjacent interior angles are equal. Find each. | 55° each |
| 14 | A regular polygon has exterior angle 30°. How many sides? | 12 sides |
| 15 | A regular polygon has interior angle 144°. How many sides? | 10 sides |
| 16 | Two parallel lines, transversal. One angle is 75°. Find all six other angles labelled around the two intersection points. | 75°, 105°, 105° at upper; 75°, 105°, 105°, 75° at lower (mirrored) |
| 17 | The interior angles of an octagon are 135°, 130°, 140°, 125°, 145°, 130°, 140°, $x$. Find $x$. | $x = 135°$ |
| 18 | A regular nonagon (9 sides) has interior angle? | 140° |
| 19 | Two parallel lines have angles labelled $a = 65°$ on the upper and $b$ co-interior on the lower. Find $b$, then find the alternate angle to $b$ on the upper line. | $b = 115°$; alternate = 115° |
| 20 | Find the angle marked $x$ in a Z-shape: two parallel lines crossed by a transversal, with $x$ alternate to $(2a + 30)°$ where $a = 25°$. | $x = 80°$ |
| 21 | In a parallel-lines diagram, one angle is $(4x - 12)°$ and a co-interior angle is $(2x + 30)°$. Find $x$. | $x = 27$ |
| 22 | A regular polygon has interior angle 150°. How many sides? Find the sum of interior angles. | 12 sides; sum 1800° |
| 23 | In the diagram, two parallel lines crossed by a transversal: acute angle on the upper line is 65°. Find the obtuse co-interior on the lower; then the acute on the lower. | Obtuse co-int = 115°; acute on lower = 65° |
| 24 | An irregular pentagon has angles 90°, 110°, 130°, $x$, $x + 20$. Find $x$. | $x = 95°$ |
| 25 | The angles around a point on a transversal between two parallel lines are $a, b, c, d$. Given $a = 70°$, find $b, c, d$. | $b = 110°, c = 70°, d = 110°$ |
| 26 | A regular polygon has interior angle 156°. Find the number of sides and verify with the sum formula. | 15 sides; sum 2340° |
| 27 | A pentagon has interior angles in the ratio 3:4:5:6:7. Find each angle. | $72°, 96°, 120°, 144°, 168°$ |
| 28 | In a regular dodecagon (12 sides), find (a) interior angle, (b) exterior angle, (c) sum of interior angles. | (a) 150° (b) 30° (c) 1800° |
| 29 | A regular hexagon and a regular triangle share a side. Find the angle at the join. | 180° (forms a straight line: 60° triangle + 120° hexagon = 180°) |
| 30 | The bisector of the exterior angle of a regular polygon meets the polygon's side at angle 75°. Find the number of sides. | 12 sides |
| 31 | Two parallel lines $\ell_1$ and $\ell_2$ are crossed by a transversal. Let $a$ be an angle on $\ell_1$ and $b$ the angle alternate to $a$ on $\ell_2$. Prove (using corresponding-angle property) that $a = b$. | See working — $a = b$ via vertically opposite + corresponding |
| 32 | A regular pentagon has its interior diagonals drawn, forming a five-pointed star. Find the angle at each "point" of the star. | 36° |
| 33 | The interior angles of an octagon are five angles each $x°$ and three angles each $(x + 10)°$. Find $x$. | $x = 131.25°$ |
| 34 | Find $x$ in a quadrilateral with angles $(x + 30)°, (2x - 10)°, (3x)°, (x + 10)°$. | $x \approx 47.1°$ |
| 35 | A regular $n$-gon has interior angle equal to $\dfrac{180(n-2)}{n}$. (a) Show that as $n \to \infty$ the interior approaches 180°. (b) For which $n$ is the interior > 150°? | (a) Limit 180° (b) $n \geq 12$ (interior 150 for $n = 12$, > 150 for $n > 12$) |
| 36 | Two parallel lines are cut by two transversals forming a quadrilateral region. The quadrilateral has two right angles. Use angle properties to prove the remaining two angles are supplementary. | Sum of all four = 360°; two right angles → remaining two = 180° (supplementary) |
| 37 | A regular polygon's interior angle is 3 times its exterior angle. Find the number of sides. | 8 sides |
| 38 | Three regular polygons meet at a point with no gap and no overlap. Each interior angle of each polygon must satisfy a constraint. Find a valid combination. | E.g. triangle (60°) + square (90°) + 12-gon (150°) — but 60+90+150 = 300, not 360. Try: hexagon (120°) × 3 → 360° ✓. |
| 39 | In a quadrilateral, the angles in order are $(x)°, (2x)°, (x - 10)°, (3x + 10)°$. Find $x$ and the angles. | $x = 51.4°$; angles ≈ 51.4°, 102.9°, 41.4°, 164.3° |
| 40 | A pentagon has 4 interior angles each $x°$ and a 5th of $(2x + 20)°$. Find $x$. | $x = 86.67°$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Two lines cross. One of the four angles is 113°. State the other three angles and name the relationship for each. | 113° (vert opp), 67°, 67° |
| 2 | A transversal crosses two parallel lines. One alternate (Z) angle is 108°. Find the other. | 108° |
| 3 | A transversal crosses two parallel lines. A corresponding (F) angle is 144°. Find the other. | 144° |
| 4 | Two co-interior (allied / "C") angles between parallel lines are 73° and $x$. Find $x$. | $x = 107°$ |
| 5 | Find the sum of the interior angles of an $n$-sided polygon with $n = 8$. | 1080° |
| 6 | Find one interior angle of a regular 10-gon. | 144° |
| 7 | Find the exterior angle of a regular 10-gon. | 36° |
| 8 | In a triangle, two angles are 80° and 60°. Find the third. | 60° |
| 9 | A pentagon has angles 100°, 110°, 95°, 105° and $x$. Find $x$. | 105° |
| 10 | A regular octagon has 8 interior angles. State the size of each. | Same. |
| 11 | Two parallel lines crossed by a transversal: the acute angle on the upper line is 41°. Find the corresponding angle on the lower line. | 41° |
| 12 | Two parallel lines have a transversal. One angle is 105°; another (alternate) is $(x + 25)$°. Find $x$. | $x = 80$ |
| 13 | The exterior angle of a triangle is 130°. The two non-adjacent interior angles are equal. Find each. | 65° each |
| 14 | A regular polygon has exterior angle 30°. How many sides? | 8 sides |
| 15 | A regular polygon has interior angle 144°. How many sides? | 12 sides |
| 16 | Two parallel lines, transversal. One angle is 75°. Find all six other angles labelled around the two intersection points. | Same with 60° / 120°. |
| 17 | The interior angles of an octagon are 135°, 130°, 140°, 125°, 145°, 130°, 140°, $x$. Find $x$. | Same. |
| 18 | A regular nonagon (9 sides) has interior angle? | ≈ 147.3° |
| 19 | Two parallel lines have angles labelled $a = 65°$ on the upper and $b$ co-interior on the lower. Find $b$, then find the alternate angle to $b$ on the upper line. | Same. |
| 20 | Find the angle marked $x$ in a Z-shape: two parallel lines crossed by a transversal, with $x$ alternate to $(2a + 30)°$ where $a = 25°$. | $x = 70°$ |
| 21 | In a parallel-lines diagram, one angle is $(4x - 12)°$ and a co-interior angle is $(2x + 30)°$. Find $x$. | $x = 36$ |
| 22 | A regular polygon has interior angle 150°. How many sides? Find the sum of interior angles. | 20 sides; sum 3240° |
| 23 | In the diagram, two parallel lines crossed by a transversal: acute angle on the upper line is 48°. Find the obtuse co-interior on the lower; then the acute on the lower. | Obtuse 132°; acute 48° |
| 24 | An irregular pentagon has angles 90°, 110°, 130°, $x$, $x + 20$. Find $x$. | $x = 117.5°$ |
| 25 | The angles around a point on a transversal between two parallel lines are $a, b, c, d$. Given $a = 70°$, find $b, c, d$. | Same. |
| 26 | A regular polygon has interior angle 156°. Find the number of sides and verify with the sum formula. | 20 sides; sum 3240° |
| 27 | A pentagon has interior angles in the ratio 3:4:5:6:7. Find each angle. | $54°, 81°, 108°, 135°, 162°$ |
| 28 | In a regular dodecagon (12 sides), find (a) interior angle, (b) exterior angle, (c) sum of interior angles. | Same. |
| 29 | A regular hexagon and a regular triangle share a side. Find the angle at the join. | 150° (90° + 60°) |
| 30 | The bisector of the exterior angle of a regular polygon meets the polygon's side at angle 75°. Find the number of sides. | Same. |
| 31 | Two parallel lines $\ell_1$ and $\ell_2$ are crossed by a transversal. Let $a$ be an angle on $\ell_1$ and $b$ the angle alternate to $a$ on $\ell_2$. Prove (using corresponding-angle property) that $a = b$. | Same. |
| 32 | A regular pentagon has its interior diagonals drawn, forming a five-pointed star. Find the angle at each "point" of the star. | Same. |
| 33 | The interior angles of an octagon are five angles each $x°$ and three angles each $(x + 10)°$. Find $x$. | $x = 140°$ |
| 34 | Find $x$ in a quadrilateral with angles $(x + 30)°, (2x - 10)°, (3x)°, (x + 10)°$. | $x = 50°$ |
| 35 | A regular $n$-gon has interior angle equal to $\dfrac{180(n-2)}{n}$. (a) Show that as $n \to \infty$ the interior approaches 180°. (b) For which $n$ is the interior > 150°? | Same. |
| 36 | Two parallel lines are cut by two transversals forming a quadrilateral region. The quadrilateral has two right angles. Use angle properties to prove the remaining two angles are supplementary. | Same. |
| 37 | A regular polygon's interior angle is 3 times its exterior angle. Find the number of sides. | 12 sides |
| 38 | Three regular polygons meet at a point with no gap and no overlap. Each interior angle of each polygon must satisfy a constraint. Find a valid combination. | Same. |
| 39 | In a quadrilateral, the angles in order are $(x)°, (2x)°, (x - 10)°, (3x + 10)°$. Find $x$ and the angles. | Same. |
| 40 | A pentagon has 4 interior angles each $x°$ and a 5th of $(2x + 20)°$. Find $x$. | $x = 78°$ |
Problèmes — Solutions détaillées
**Angles in parallel lines with algebra.** Two parallel lines are crossed by a transversal. One marked angle is $(3x + 20)°$ and the co-interior angle on the same side is $(2x + 40)°$. (a) Set up and solve an equation in $x$. (b) State the size of both angles and verify they are co-interior. (c) Find the corresponding angle of the $(3x + 20)°$ angle on the other parallel line.
(a) $5x + 60 = 180, x = 24$ (b) 92° and 88°; sum 180° ✓ (c) 92°
**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) 18° (b) 20 sides (c) 3240°
**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) 108° (b) 36° (c) $5 \times 36° = 180°$
**Two-step parallel-line problem.** In a diagram with two parallel lines $\ell_1$ and $\ell_2$, and a transversal: - The acute angle at $\ell_1$ is $(2x + 10)°$. - The vertically opposite angle is $(3x - 20)°$. (a) Use vertically-opposite-angles property to find $x$. (b) Find both angles. (c) Find the obtuse co-interior angle on $\ell_2$.
(a) $x = 30$ (b) 70° each (c) 110°
**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) Pentagon 108°, triangle 60° (b) 168° (c) No — 168° ≠ 360° per pair
**Sum-of-angles identity.** Prove that the sum of the exterior angles of any convex polygon equals 360°.
Each interior + ext = 180°. Sum of interiors $= (n-2) \times 180$. Sum of all interior + ext = $n \times 180$. So sum of exterior = $n \times 180 - (n-2) \times 180 = 2 \times 180 = 360°$ ✓
**Regular hexagon investigation.** A regular hexagon has side $a$. (a) Find the interior angle. (b) The hexagon can be split into 6 equilateral triangles. Use this to find the area in terms of $a$. (c) Find the perimeter.
(a) 120° (b) $\tfrac{3\sqrt{3}}{2} a^2$ (c) $6a$
**Mixed angles in a complex figure.** In a figure: two parallel lines crossed by a transversal create angles labelled $a, b, c, d$ at the upper intersection and $e, f, g, h$ at the lower. Given $a = 60°$: (a) Find $b, c, d$ at the upper intersection. (b) Find $e, f, g, h$ at the lower intersection. (c) Identify which are alternate, corresponding, co-interior, vertically opposite.
(a) $b = 120°, c = 60°, d = 120°$ (b) $e = 60°, f = 120°, g = 60°, h = 120°$ (c) See working
**Irregular polygon angles.** A pentagon has angles 100°, 110°, 95°, 130°, $x$. (a) Find $x$. (b) Classify the polygon (convex or concave). (c) If $x > 180°$, what would that mean?
(a) $x = 105°$ (b) Convex (all < 180°) (c) Concave
**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) 60, 90, 120 all divide 360 (b) 108° does not (c) E.g. an irregular pentagon with specific angle measures
**Parallel-line angle proof.** In a diagram, parallel lines $\ell_1$ and $\ell_2$ are crossed by transversal $t$. Show that the sum of the two co-interior angles is 180°.
Use the fact that $a + b + c + d = 360°$ at the intersection and use alternate / vertically opposite identities.
**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 $n + 5$ sides where this polygon has $n$ sides is bigger.
(a) 10 (b) 1440° (c) 36° (d) Interior increases with $n$