Angles at points, lines and parallel lines

(a) $63$°

1. $$180-41-76$$
2. Angles on a straight line add to $180^\circ.$
3. x = 180 $-$ 41 $-$ 76 = 63.

• P1 Establishing $180-41-76$ or an equivalent valid method.
• A1 Correct answer: $63$°

(a) $128^\circ$

1. $a$ is vertically opposite the $128^\circ$ angle, so it is equal to it.

• B1 Correct answer: $128^\circ$.

(b) Vertically opposite angles are equal.

1. Two crossing straight lines make two pairs of vertically opposite angles, and each pair is equal.

• C1 Choosing "vertically opposite angles are equal".

(c) $52^\circ$

1. Angles on a straight line add up to $180^\circ$, so $b = 180 - 128 = 52^\circ$.

• B1 Correct answer: $52^\circ$.

(a) $135$°

1. Adjacent angles on a straight line sum to 180 degrees.
   $$180-45$$
2. Therefore $135$°.

• M1 Adjacent angles on a straight line sum to 180 degrees.
• A1 Correct answer: $135$°

(b) It is $45^\circ$ because corresponding angles between parallel lines are equal.

1. It is $45^\circ$ because corresponding angles between parallel lines are equal.

• C1 Correct conclusion with supporting reasoning: It is $45^\circ$ because corresponding angles between parallel lines are equal.

(a) $x = 36$

1. The four angles form one full turn.
   $$x+2x+10+3x+4x-10=360$$
2. Collect the coefficients before dividing.
   $$10x=360$$
3. Therefore $x = 36$.

• M1 The four angles form one full turn.
• M1 Collect the coefficients before dividing.
• A1 Correct answer: $x = 36$

(b) $134$°

1. Substitute x into the largest coefficient expression and compare.
   $$4\times (36)-10$$
2. Therefore $134$°.

• M1 Substitute x into the largest coefficient expression and compare.
• A1 Correct answer: $134$°

(a) $\frac{43}{2}$

1. Co-interior angles between parallel lines are supplementary.
   $$3x+4+5x+4=180$$
2. Therefore $\frac{43}{2}$.

• M1 Co-interior angles between parallel lines are supplementary.
• A1 Correct answer: $\frac{43}{2}$

(b) $68.5$°

1. Substitute into the smaller angle expression.
   $$3\times (43/2)+4$$
2. Therefore $68.5$°.

• M1 Substitute into the smaller angle expression.
• A1 Correct answer: $68.5$°

(a) $20$

1. Alternate angles are equal, so form an equation.
   $$3y=y+40$$
2. Therefore $20$.

• M1 Alternate angles are equal, so form an equation.
• A1 Correct answer: $20$

(b) $60$°

1. Substitute y into 3y.
   $$3\times 20$$
2. Therefore $60$°.

• M1 Substitute y into 3y.
• A1 Correct answer: $60$°

(c) Alternate angles between parallel lines are equal.

1. Alternate angles between parallel lines are equal.

• C1 Correct conclusion with supporting reasoning: Alternate angles between parallel lines are equal.

(a) $16$

1. Vertically opposite angles are equal.
   $$4x+12=6x-20$$
2. Therefore $16$.

• M1 Vertically opposite angles are equal.
• A1 Correct answer: $16$

(b) $104$°

1. The other angles lie on a straight line with a 76° angle.
   $$180-76$$
2. Therefore $104$°.

• M1 The other angles lie on a straight line with a 76° angle.
• A1 Correct answer: $104$°

(a) No: $x = 24$, so the corresponding angles are $80^\circ$ and $78^\circ$, which are not equal.

1. Angles on a straight line add up to $180^\circ$.
   $$(5x - 20) + (3x + 8) = 180$$
2. Solve.
   $$8x - 12 = 180 \quad\Rightarrow\quad x = 24$$
3. At $G$ the angle is $3(24) + 8 = 80^\circ$. At $H$ the corresponding angle is $2(24) + 30 = 78^\circ$.
4. Corresponding angles on parallel lines are equal. $80^\circ \ne 78^\circ$, so $AB$ is not parallel to $CD$.

• P1 Forming the equation from angles on a straight line.
• P1 Solving to $x = 24$.
• P1 Working out both corresponding angles, $80^\circ$ and $78^\circ$.
• C1 "Not parallel", because the corresponding angles ($80^\circ$ and $78^\circ$) are not equal.