Triangles, interior and exterior polygon angles

(a) $36^\circ$

1. The exterior angles of any polygon add up to $360^\circ$.
2. $$360 \div 10 = 36$$

• M1 Dividing $360$ by $10$.
• A1 Correct answer: $36^\circ$.

(b) $144^\circ$

1. An interior and an exterior angle sit on a straight line, so they add up to $180^\circ$.
2. $$180 - 36 = 144$$

• B1 Correct answer: $144^\circ$.

(a) $68$°

1. $$2(3x+5)+4x=180$$
2. $$10x=170$$
3. The base angles ABC and BCA are equal because AB = AC.
4. Angles in a triangle sum to $180^\circ$: 2(3x + 5) + 4x = 180.
5. 10x + 10 = 180, so x = 17 and angle BAC = 4 $\times$ 17 = $68^\circ.$

• P1 Establishing $2(3x+5)+4x=180$ or an equivalent valid method.
• P1 Establishing $10x=170$ or an equivalent valid method.
• A1 Correct answer: $68$°

(a) $16$

1. $$180/8$$
2. $$360/22.5$$
3. An interior angle and its exterior angle total $180^\circ.$
4. If the exterior angle is $e^\circ$, then 7e + e = 180, giving e = 22.5.
5. Exterior angles total $360^\circ$, so the number of sides is 360 $\div$ 22.5 = 16.

• P1 Establishing $180/8$ or an equivalent valid method.
• P1 Establishing $360/22.5$ or an equivalent valid method.
• A1 Correct answer: $16$

(a) $45$°

1. The exterior angles make a full turn and are all equal.
   $$360/8$$
2. Therefore $45$°.

• M1 The exterior angles make a full turn and are all equal.
• A1 Correct answer: $45$°

(b) $1080$°

1. Split the polygon into two fewer triangles than sides.
   $$(8-2)\times 180$$
2. Therefore $1080$°.

• M1 Split the polygon into two fewer triangles than sides.
• A1 Correct answer: $1080$°

(a) $9$

1. Find the supplementary exterior angle.
   $$180-(140)$$
2. Divide a full turn by the exterior angle.
   $$360/(40)$$
3. Therefore $9$.

• M1 Find the supplementary exterior angle.
• M1 Divide a full turn by the exterior angle.
• A1 Correct answer: $9$

(b) A polygon has a whole number of edges; a non-integer result would mean the proposed regular polygon cannot exist.

1. A polygon has a whole number of edges; a non-integer result would mean the proposed regular polygon cannot exist.

• C1 Correct conclusion with supporting reasoning: A polygon has a whole number of edges; a non-integer result would mean the proposed regular polygon cannot exist.

(a) $106$°

1. The interior angle at C is supplementary to the exterior angle.
   $$180-143$$
2. Both base angles equal this angle.
   $$180-2\times 37$$
3. Therefore $106$°.

• M1 The interior angle at C is supplementary to the exterior angle.
• M1 Both base angles equal this angle.
• A1 Correct answer: $106$°

(b) Angles on a straight line sum to 180 degrees; equal sides have equal opposite angles; a triangle’s angles sum to 180 degrees.

1. Angles on a straight line sum to 180 degrees; equal sides have equal opposite angles; a triangle’s angles sum to 180 degrees.

• C1 Correct conclusion with supporting reasoning: Angles on a straight line sum to 180 degrees; equal sides have equal opposite angles; a triangle’s angles sum to 180 degrees.

(a) $70$

1. Find the sum of the interior angles of a pentagon.
   $$(5-2)\times 180$$
2. Form and solve an equation.
   $$3x+330=540$$
3. Therefore $70$.

• M1 Find the sum of the interior angles of a pentagon.
• M1 Form and solve an equation.
• A1 Correct answer: $70$

(b) No. Its angles are 70°, 140°, 100°, 110° and 120°, which are not all equal.

1. No. Its angles are 70°, 140°, 100°, 110° and 120°, which are not all equal.

• C1 Correct conclusion with supporting reasoning: No. Its angles are 70°, 140°, 100°, 110° and 120°, which are not all equal.

(a) 15

1. Each exterior angle is $180^\circ - 156^\circ = 24^\circ$.
2. The exterior angles add up to $360^\circ$.
   $$360 \div 24 = 15$$

• P1 Finding the exterior angle, $24^\circ$.
• P1 Dividing $360$ by their exterior angle.
• A1 Correct answer: 15 sides.

(b) $2340^\circ$

1. A polygon with $n$ sides splits into $n - 2$ triangles from one vertex.
2. $$(15 - 2) \times 180 = 2340$$

• M1 $(n - 2) \times 180$ with their $n$, or $n \times 156$.
• A1 Correct answer: $2340^\circ$.