Hard GCSE Angles in parallel lines Questions

Write down what you are given

Angle $APQ$ is $43^{\circ}$. The question is which angle fact takes you from there to the angle that was written down.

Look at the first step, which is not the one being asked about

The first step uses a different fact (alternate angles are equal) to get angle $APQ = 43^{\circ}$. The question asks about the SECOND step only.

Describe where angle $APQ$ and angle $BPE$ sit

Angle $APQ$ and angle $BPE$ are opposite each other at the point where two straight lines cross.

Name the fact

Vertically opposite angles are equal.

Check the fact really does give that answer

Using the fact on angle $APQ$ gives $43^{\circ}$, which is exactly what was written down.

Check the numbers in the whole chain

Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

Rule out one of the other answers

"Alternate angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle $APQ$ and angle $BPE$ are not that kind of pair, so it would earn no mark.

Rule out one of the other answers

"Corresponding angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle $APQ$ and angle $BPE$ are not that kind of pair, so it would earn no mark.

Rule out one of the other answers

"Co-interior angles add up to $180^{\circ}$." is a true fact, but it does not describe this pair: applying it here would give $137^{\circ}$, not $43^{\circ}$.

Rule out one of the other answers

"Vertically opposite angles add up to $180^{\circ}$." is simply not true, so it cannot justify anything. Used here it would give $137^{\circ}$.

Compare the two steps

The two steps use different facts. Naming the first one for the second step is the commonest error here.

Check the fact against the figure once more

Angle $APQ$ and angle $BPE$ are opposite each other at the point where two straight lines cross.

Write the reason out in full

A reason like "parallel lines" on its own earns nothing: the fact has to be named. Vertically opposite angles are equal.

Note the mark scheme

One mark is available and it is for naming the correct angle fact.

State the fact that earns the mark

The reason that must be written down is: Vertically opposite angles are equal.

Write down what you are given

Angle $QPB$ is $74^{\circ}$. The question is which angle fact takes you from there to the angle that was written down.

Look at the first step, which is not the one being asked about

The first step uses a different fact (angles on a straight line add up to $180^{\circ}$) to get angle $QPB = 74^{\circ}$. The question asks about the SECOND step only.

Describe where angle $QPB$ and angle $PQC$ sit

Angle $QPB$ and angle $PQC$ lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

Name the fact

Alternate angles are equal.

Check the fact really does give that answer

Using the fact on angle $QPB$ gives $74^{\circ}$, which is exactly what was written down.

Check the numbers in the whole chain

Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

Rule out one of the other answers

"Angles on a straight line add up to $180^{\circ}$." is a true fact, but it does not describe this pair: applying it here would give $106^{\circ}$, not $74^{\circ}$.

Rule out one of the other answers

"Co-interior angles add up to $180^{\circ}$." is a true fact, but it does not describe this pair: applying it here would give $106^{\circ}$, not $74^{\circ}$.

Rule out one of the other answers

"Vertically opposite angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle $QPB$ and angle $PQC$ are not that kind of pair, so it would earn no mark.

Rule out one of the other answers

"Alternate angles add up to $180^{\circ}$." is simply not true, so it cannot justify anything. Used here it would give $106^{\circ}$.

Compare the two steps

The two steps use different facts. Naming the first one for the second step is the commonest error here.

Check the fact against the figure once more

Angle $QPB$ and angle $PQC$ lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

Write the reason out in full

A reason like "parallel lines" on its own earns nothing: the fact has to be named. Alternate angles are equal.

Note the mark scheme

One mark is available and it is for naming the correct angle fact.

State the fact that earns the mark

The reason that must be written down is: Alternate angles are equal.

Write down what you are given

Angle $QPB$ is $59^{\circ}$. The question is which angle fact takes you from there to the angle that was written down.

Look at the first step, which is not the one being asked about

The first step uses a different fact (vertically opposite angles are equal) to get angle $QPB = 59^{\circ}$. The question asks about the SECOND step only.

Describe where angle $QPB$ and angle $PQD$ sit

Angle $QPB$ and angle $PQD$ lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

Name the fact

Co-interior angles add up to $180^{\circ}$.

Check the fact really does give that answer

Using the fact on angle $QPB$ gives $121^{\circ}$, which is exactly what was written down.

Check the numbers in the whole chain

Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

Rule out one of the other answers

"Vertically opposite angles are equal." is a true fact, but it does not describe this pair: applying it here would give $59^{\circ}$, not $121^{\circ}$.

Rule out one of the other answers

"Alternate angles are equal." is a true fact, but it does not describe this pair: applying it here would give $59^{\circ}$, not $121^{\circ}$.

Rule out one of the other answers

"Corresponding angles are equal." is a true fact, but it does not describe this pair: applying it here would give $59^{\circ}$, not $121^{\circ}$.

Rule out one of the other answers

"Co-interior angles are equal." is simply not true, so it cannot justify anything. Used here it would give $59^{\circ}$.

Compare the two steps

The two steps use different facts. Naming the first one for the second step is the commonest error here.

Check the fact against the figure once more

Angle $QPB$ and angle $PQD$ lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

Write the reason out in full

A reason like "parallel lines" on its own earns nothing: the fact has to be named. Co-interior angles add up to $180^{\circ}$.

Note the mark scheme

One mark is available and it is for naming the correct angle fact.

State the fact that earns the mark

The reason that must be written down is: Co-interior angles add up to $180^{\circ}$.

Write down the two algebraic angles

Both angles are written in terms of $x$. One angle fact will turn that into an equation.

Name the fact that links angle $QPB$ and angle $DQF$

Corresponding angles are equal. Angle $QPB$ and angle $DQF$ sit in matching positions at the two crossings — the F-shape — so they are corresponding angles.

Form an equation

Corresponding angles are equal. So the two expressions are equal, which gives an equation in $x$.

Collect the like terms

Gather the $x$ terms on one side and the numbers on the other.

Solve for $x$

Dividing both sides by $-2$ gives $x = 15$.

Substitute $x$ back to find angle $QPB$

Putting $x = 15$ into the first expression gives $43^\circ$.

Substitute $x$ back to find angle $DQF$

Putting $x = 15$ into the second expression gives $43^\circ$.

Check the two angles against the fact

Corresponding angles are equal. The two angles came out the same size, so $x$ is right.

Name the fact that links angle $QPB$ and angle $PQD$

Co-interior angles add up to $180^{\circ}$. Angle $QPB$ and angle $PQD$ lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

Work out angle $PQD$

Co-interior angles add up to $180^{\circ}$. Taking angle $QPB$, $43^{\circ}$, away from $180^{\circ}$ leaves $137^{\circ}$.

Check both angles are possible

Both angles came out between $0^{\circ}$ and $180^{\circ}$ ($43^{\circ}$ and $43^{\circ}$), so the figure can really be drawn.

Avoid the usual mistake

These angles are equal, not supplementary. Setting the two expressions to add up to $180$ is the classic error and gives the wrong value of $x$.

Write down the reasons that earn the marks

In the exam the marks are for the reasons as much as for the numbers, so every step must name the angle fact it uses: corresponding angles are equal, then co-interior angles add up to $180^{\circ}$.

Read the answer back into the figure

Put the numbers back on the diagram and check each angle looks the size it should — acute angles acute, obtuse angles obtuse.

State the size of angle $PQD$

Angle $PQD$ measures $137^{\circ}$.

Write down the two algebraic angles

Both angles are written in terms of $x$. One angle fact will turn that into an equation.

Name the fact that links angle $QPB$ and angle $PQC$

Alternate angles are equal. Angle $QPB$ and angle $PQC$ lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

Form an equation

Alternate angles are equal. So the two expressions are equal, which gives an equation in $x$.

Collect the like terms

Gather the $x$ terms on one side and the numbers on the other.

Solve for $x$

Dividing both sides by $-2$ gives $x = 10$.

Substitute $x$ back to find angle $QPB$

Putting $x = 10$ into the first expression gives $47^\circ$.

Substitute $x$ back to find angle $PQC$

Putting $x = 10$ into the second expression gives $47^\circ$.

Check the two angles against the fact

Alternate angles are equal. The two angles came out the same size, so $x$ is right.

Name the fact that links angle $QPB$ and angle $BPE$

Angles on a straight line add up to $180^{\circ}$. Angle $QPB$ and angle $BPE$ are next to each other on a straight line, so together they make a half turn.

Work out angle $BPE$

Angles on a straight line add up to $180^{\circ}$. Taking angle $QPB$, $47^{\circ}$, away from $180^{\circ}$ leaves $133^{\circ}$.

Check both angles are possible

Both angles came out between $0^{\circ}$ and $180^{\circ}$ ($47^{\circ}$ and $47^{\circ}$), so the figure can really be drawn.

Avoid the usual mistake

These angles are equal, not supplementary. Setting the two expressions to add up to $180$ is the classic error and gives the wrong value of $x$.

Write down the reasons that earn the marks

In the exam the marks are for the reasons as much as for the numbers, so every step must name the angle fact it uses: alternate angles are equal, then angles on a straight line add up to $180^{\circ}$.

Read the answer back into the figure

Put the numbers back on the diagram and check each angle looks the size it should — acute angles acute, obtuse angles obtuse.

State the size of angle $BPE$

Angle $BPE$ measures $133^{\circ}$.