Sec 1 Angles & Parallel Lines: Methods and Examples

Secondary 1 Mathematics · G3 focus · Study guide

Recognise angle relationships, give geometric reasons and solve unknown angles with a parallel-line explorer.

Concept → worked examples → practice14 min read · practice linked below

Read the angle before choosing a rule

An angle measures a turn between two rays. In $\angle ABC$ , the middle letter $B$ is the vertex: the rays are $BA$ and $BC$ . A small square marks a right angle; matching arrow marks identify parallel lines. A sketch may not be drawn to scale, so calculate from the given facts rather than measuring the picture.

Names describe the size of the turn
Type	Size
Acute	Between $0^\circ$ and $90^\circ$
Right	Exactly $90^\circ$
Obtuse	Between $90^\circ$ and $180^\circ$
Straight	Exactly $180^\circ$
Reflex	Between $180^\circ$ and $360^\circ$
Full turn	Exactly $360^\circ$

Two angles are complementary when they total $90^\circ$ , and supplementary when they total $180^\circ$ . These words describe their sum; the angles need not be next to each other.

Worked example 1: distinguish the smaller and reflex angles

Two rays form a smaller angle of $74^\circ$ . Find the reflex angle between the same rays.

A full turn around the vertex is $360^\circ$ . The smaller turn and the reflex turn make that full turn.
Subtract the smaller angle: $\text{reflex angle}=360^\circ-74^\circ=\boxed{286^\circ}.$

Check: $74+286=360$ , and $286^\circ$ is between $180^\circ$ and $360^\circ$ , as a reflex angle must be. Subtracting from $180^\circ$ would find the supplementary angle, which is a different quantity.

Three facts that do not need parallel lines

Adjacent angles that fill a straight line add to $180^\circ$ .
Non-overlapping angles that fill one turn around a point add to $360^\circ$ .
When two straight lines cross, vertically opposite angles are equal. They face each other across the intersection and share no side.

The two lines continue straight through

O

. Equality follows from the line geometry, not the orientation of the page.

Worked example 2: turn a straight-line fact into an equation

Two adjacent angles on a straight line are $(3x+18)^\circ$ and $(2x+12)^\circ$ . Find $x$ and both angles.

State the reason, then form the equation: $(3x+18)+(2x+12)=180\quad\text{(angles on a straight line)}.$
Collect terms and solve: $5x+30=180,\qquad5x=150,\qquad x=30.$
Substitute into each expression: $3(30)+18=\boxed{108^\circ},\qquad2(30)+12=\boxed{72^\circ}.$

Check: $108+72=180$ . The answer $x=30$ is the unknown number, not automatically either requested angle.

Worked example 3: work around a point

Four non-overlapping angles around a point are a right angle, $85^\circ$ , $110^\circ$ and $p^\circ$ . Find $p$ .

Replace the right-angle mark by $90^\circ$ . All four angles together fill one complete turn.
Use angles at a point: $90+85+110+p=360.$
The known angles total $285^\circ$ , so $p=360-285=\boxed{75}.$

Answer: the missing angle is $75^\circ$ . Check: $90+85+110+75=360$ . Do not use $180^\circ$ unless the selected angles fill a straight line.

A transversal crosses two parallel lines

A transversal is a line that crosses two other lines. When those two lines are parallel:

Corresponding angles occupy the same relative position at the two intersections and are equal.
Alternate interior angles lie between the parallel lines, on opposite sides of the transversal, and are equal.
Co-interior angles lie between the parallel lines, on the same side of the transversal, and add to $180^\circ$ .

“F”, “Z” and “C” shapes can help you spot the pairs, but they may be rotated or reflected. Check the positions and state the angle reason. Without parallel lines, these equalities and sums are not guaranteed.

Explore which pairs stay equal

At the upper crossing the angles are A, B, C, D clockwise from the top right. At the lower crossing they are E, F, G, H in the same order. Start with A = 60°: E and C are also 60°, while B is 120°.

Each letter stays in its labelled position. The angle beside it is recalculated as the transversal turns.

A = E = 60°. Corresponding angles are equal because the lines are parallel.

Worked example 4: chain two angle reasons

Use the letter positions in the diagram above. Suppose $A=68^\circ$ . Find $E$ , $C$ and $B$ .

$A$ and $E$ are in the same relative position: $E=\boxed{68^\circ}\quad\text{(corresponding angles, parallel lines)}.$
$C$ and $E$ are between the lines on opposite sides of the transversal: $C=\boxed{68^\circ}\quad\text{(alternate interior angles, parallel lines)}.$
$B$ and $E$ are between the lines on the same side: $B+68^\circ=180^\circ,\qquad B=\boxed{112^\circ}\quad\text{(co-interior angles)}.$

Check another way: $C=A=68^\circ$ by vertically opposite angles; $A+B=180^\circ$ at the upper crossing. Both routes agree. Move the explorer to 68° to see the pair positions.

Worked example 5: parallel-line angles with algebra

Two co-interior angles between parallel lines are $(4x+8)^\circ$ and $(2x+28)^\circ$ . Find both angles.

Co-interior angles are supplementary, so $(4x+8)+(2x+28)=180.$
Solve: $6x+36=180$ , hence $6x=144$ and $x=24$ .
Evaluate the original expressions: $4(24)+8=\boxed{104^\circ},\qquad2(24)+28=\boxed{76^\circ}.$

Check: both angles lie between $0^\circ$ and $180^\circ$ , and $104+76=180$ . Setting the two expressions equal would use the wrong relationship.

Common mistakes and useful checks

Match the rule to the geometry
Mistake	Repair
Assuming two lines are parallel because they look parallel	Use the given statement or matching arrow marks. A picture alone does not establish parallelism.
Making co-interior angles equal	They total $180^\circ$ . Corresponding and alternate pairs are the equal pairs.
Calling adjacent angles vertically opposite	Vertically opposite angles share only their vertex, not a side.
Giving $x$ when the question asks for an angle expression	Substitute $x$ back and include the degree sign in the angle answer.
Counting an angle twice around a point	Choose non-overlapping sectors that cover exactly one full turn.

For a longer diagram, write each result next to its reason and mark it on the figure before continuing. Add an auxiliary line parallel to the given lines only when it helps you form a recognised pair.

Practise identifying the reason

Use our Sec 1 angles and parallel lines question bank. Attempt each question on paper and write an angle reason beside every step before checking the worked solution. Continue with triangles, quadrilaterals and polygons to use these facts inside shapes.