Angles and parallel lines questions with answers

Questions

1. Question 1Alternate angles

   In the diagram, $AB$ is parallel to $CD$, and the straight line $EF$ cuts them at $P$ and $Q$. Find $x$.

   Show the worked solution

   1. Step 1: Find x and give the reason$x = 42$ (alt. ∠s, $AB \parallel CD$)
2. Question 2Corresponding angles

   In the diagram, $AB$ is parallel to $CD$, and the straight line $EF$ cuts them at $P$ and $Q$. Find $x$.

   Show the worked solution

   1. Step 1: Find x and give the reason$x = 61$ (corr. ∠s, $AB \parallel CD$)
3. Question 3Co-interior angles

   In the diagram, $AB$ is parallel to $CD$, and the straight line $EF$ cuts them at $P$ and $Q$. Find $x$.

   Show the worked solution

   1. Step 1: Which angle fact links x° and the given angle?$x^\circ + 137^\circ = 180^\circ$ (int. ∠s, $AB \parallel CD$)
   2. Step 2: Find x$x = 180 - 137$$x = 43$
4. Question 4Two steps with a transversal

   In the diagram, $AB$ is parallel to $CD$, and the straight line $EF$ cuts them at $P$ and $Q$. Find $x$.

   Show the worked solution

   1. Step 1: First carry the given angle across to P$\angle QPB = 119^\circ$ (corr. ∠s, $AB \parallel CD$)
   2. Step 2: Then use it to find x$x = 119$ (vert. opp. ∠s)
5. Question 5Parallel lines with expressions

   In the diagram, $AB$ is parallel to $CD$, and the straight line $EF$ cuts them at $P$ and $Q$. Find $x$.

   Show the worked solution

   1. Step 1: Write an equation in x$3x - 10 = x + 30$ (corr. ∠s, $AB \parallel CD$)
   2. Step 2: Collect the x terms on one side and the numbers on the other$3x - x = 30 + 10$$2x = 40$$x = 20$
6. Question 6A bend between parallel lines (construction line)

   In the diagram, $AB$ is parallel to $CD$. Find $x$.

   Show the worked solution

   1. Step 1: Draw a line through E parallel to AB and CDDraw the line $EF$ through $E$, parallel to $AB$ and $CD$, pointing towards $A$ and $C$. It splits $x^\circ$ into $\angle BEF$ and $\angle FED$.
   2. Step 2: The part next to AB$\angle BEF = 180^\circ - 127^\circ$$\angle BEF = 53^\circ$ (int. ∠s, $AB \parallel EF$)
   3. Step 3: The part next to CD$\angle FED = 180^\circ - 131^\circ$$\angle FED = 49^\circ$ (int. ∠s, $CD \parallel EF$)
   4. Step 4: Add the two parts$x = 53 + 49$$x = 102$
7. Question 7A bend with a reflex angle as an expression

   In the diagram, $AB$ is parallel to $CD$. The reflex angle $BED$ is $(5x - 10)^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Draw a line through E parallel to AB and CDDraw the line $EF$ through $E$, parallel to $AB$ and $CD$, pointing towards $B$ and $D$. It splits the angle $BED$ inside the bend into $\angle BEF$ and $\angle FED$.
   2. Step 2: The part next to AB$\angle BEF = (x + 15)^\circ$ (alt. ∠s, $AB \parallel EF$)
   3. Step 3: The part next to CD$\angle FED = 31^\circ$ (alt. ∠s, $CD \parallel EF$)
   4. Step 4: The reflex angle and the two parts go all the way round E$(5x - 10)^\circ + (x + 15)^\circ + 31^\circ = 360^\circ$ (∠s at a pt.)
   5. Step 5: Collect like terms and solve$6x + 36 = 360$$6x = 324$$x = 54$
8. Question 8Isosceles triangles and the longest side

   In the diagram, $AB$ is parallel to $CD$. Which side of triangle $PQS$ is the longest? Choose the answer.

   Show the worked solution

   1. Step 1: Find ∠PQS$\angle PQS = \angle APQ$$\phantom{\angle PQS} = 63^\circ$ (alt. ∠s, $AB \parallel CD$)
   2. Step 2: Find ∠PSQ$\angle PSQ = \angle BPS$$\phantom{\angle PSQ} = 34^\circ$ (alt. ∠s, $AB \parallel CD$)
   3. Step 3: Find ∠QPS$\angle QPS = 180^\circ - 63^\circ - 34^\circ$$\phantom{\angle QPS} = 83^\circ$ (∠s on a str. line)
   4. Step 4: The longest side is opposite the largest angleThe largest angle is $83^\circ$, at $P$, so the longest side is $QS$, opposite it.