Sec 1 Maths · Chapter 10

Basic Geometry

Learn to name and measure angles, use the properties of intersecting lines, and reason with parallel lines and a transversal.

This classroom resource pairs precise, clean diagrams with step-by-step explanations. Every angle is named with the vertex in the middle: ∠ABC means the angle at B.

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Photorealistic study desk with protractor, ruler, pencil labelled Basic Geometry and intersecting lines for Sec 1 Maths

Learning goals

Read diagrams

Identify points, rays, lines, vertices, and angles using correct notation.

Explain angle rules

Use a straight line, a full turn, vertically opposite angles, and parallel-line properties.

Solve problems

Combine properties in two- or three-step questions and explain the reasoning.

10.1 Basic geometrical concepts and notations

1. Naming an angle

A line extends indefinitely in both directions. A ray has one endpoint and extends in one direction. An angle is formed by two rays meeting at a vertex.

Line AB

AB

Extends indefinitely in both directions.

Ray CD

CD

Starts at C and extends through D.

Line segment EF

EF

Has two endpoints, E and F.

BAC

How to read ∠ABC

The middle letter B identifies the vertex. The arms are BA and BC.

∠ABC = ∠CBA because both names identify the angle between BA and BC.

∠BAC is a different angle: its vertex is A.

Angle types

TypeMeasureExample
AcuteBetween 0° and 90°45°
Right90°90°
ObtuseBetween 90° and 180°120°
Straight180°180°
ReflexBetween 180° and 360°250°
Full turn360°360°
Teacher reminder: A diagram may not be drawn to scale. Students should use the stated angles and known properties, not estimate by eye.

2. Measuring and drawing angles

Use a physical semicircular protractor with two scales. The scale you read depends on which ray is aligned with 0°.

Transparent semicircular school protractor with inner and outer degree scales from 0° to 180°

School-style protractor: align its centre with the angle vertex and select the scale that starts at 0° along the chosen arm.

Worked method: measuring 60°

  1. Put the centre hole on the vertex.
  2. Align the starting ray with the 0° mark on the appropriate scale.
  3. For a 60° angle, the other ray should meet the 60° mark on that same scale.
  4. Check that 60° is acute; if you read 120°, you probably used the opposite scale.

To draw 60°, mark 60° on that scale, then draw a ray from the vertex through the mark.

  1. Put the protractor centre precisely on the vertex.
  2. Align the 0° baseline with one arm of the angle.
  3. Check whether the angle is acute or obtuse before choosing the correct inner or outer scale.
  4. Read the value where the other arm crosses the scale. Record the answer in degrees.
Complementary angles add to 90°. Supplementary angles add to 180°.

Example: Complementary angles

BACD

Given ∠ABD = 34° and ∠ABC = 90°, find ∠DBC.

Reveal working

∠DBC = 90° − 34° = 56° (complementary angles).

Example: Supplementary angles

PQSR

Given ∠PQR = 117° and ∠RQS is its supplement, find ∠RQS.

Reveal working

∠RQS = 180° − 117° = 63°.

10.2 Angles formed by intersecting lines

3. Straight lines, angles at a point and vertically opposite angles

ABCDO

Three rules to teach

Angles on a straight line: adjacent angles sum to 180°.

Angles at a point: a complete turn sums to 360°.

Vertically opposite angles: opposite angles formed by intersecting straight lines are equal.

For the diagram: ∠AOB = ∠COD and ∠AOC = ∠BOD.

Worked example: Vertically opposite

In the figure, lines AOD and BOC intersect at O. Given ∠AOB = 112°, find ∠COD and ∠BOD.

Reveal step-by-step solution

∠COD = 112° (vertically opposite angles).
∠BOD = 180° − 112° = 68° (angles on a straight line).

Worked example: Angles at a point

OABC

Given ∠AOB = 130° and ∠BOC = 95°, find the remaining angle ∠COA (the angle from OC to OA that completes the full turn).

Show working

∠COA = 360° − 130° − 95° = 135° (angles at a point).

Worked example: Algebra

A, O and C lie on a straight line, and ray OB lies between OA and OC. Given ∠AOB = (3x + 10)° and ∠BOC = (2x + 20)°, find x.

Reveal step-by-step solution

(3x + 10) + (2x + 20) = 180.
5x + 30 = 180.
5x = 150, so x = 30.

10.3 Parallel lines and a transversal

4. Corresponding, alternate and interior angles

A transversal is a line that intersects two or more other lines. When it crosses parallel lines, specific angle relationships follow.

ATB CUD EF

Three key relationships

Corresponding: angles in matching positions are equal.

Alternate: angles on opposite sides of the transversal, between the parallel lines, are equal.

Interior on the same side: their sum is 180°.

In the figure, AB ∥ CD. The matching arrows indicate the parallel lines.

Corresponding

Given ∠BTU = 52°, find ∠DUF.

Show working

∠DUF = 52° (corresponding angles).

Alternate angles

Given ∠ATU = 128°, find ∠TUD.

Show working

∠TUD = 128° (alternate angles).

Interior angles

Given ∠BTU = 52°, find ∠TUD.

Show working

∠TUD = 180° − 52° = 128° (interior angles on the same side).

Interactive classroom demonstration: choose a property and change the transversal angle.

Explore angles in parallel lines

Choose an angle relationship, then drag the blue handle or use the slider. The two green lines remain parallel while the transversal changes direction.

First angleSecond angle

The blue handle can be moved up or down. All point names update with the diagram.

Select a property

30°–150°

Worked example: Algebra with corresponding angles

Using the labelled parallel-line figure above, AB ∥ CD. Given ∠BTU = (2𝑥 + 14)° and ∠DUF = (3𝑥 − 9)°, find 𝑥 and ∠BTU.

Show working

2𝑥 + 14 = 3𝑥 − 9 (corresponding angles).
𝑥 = 23.
∠BTU = 2 × 23 + 14 = 60°.

Common trap: These rules require the lines to be parallel. Matching arrow marks establish that condition; do not assume lines are parallel just because they look that way.

5. Multi-step geometry reasoning

Students should identify the unknown angle, choose the relevant properties and write the reason for each step. A useful advanced technique is to draw an auxiliary line parallel to the given pair.

Worked example: Zigzag between parallel lines

APB CRDQ

In the figure, AB ∥ CD. Given ∠BPQ = 36° and ∠DRQ = 42°, find ∠PQR.

Step-by-step reasoning
  1. Draw a line through Q parallel to AB and CD.
  2. By alternate angles, the part of ∠PQR above the auxiliary line is 36°.
  3. By alternate angles, the part below it is 42°.
  4. ∠PQR = 36° + 42° = 78°.

Worked example: Triangle and intersecting lines

ABCD

In △ABC, given ∠BAC = 47° and ∠ABC = 63°, find ∠ACB and ∠CBD, where AB is extended beyond B to D.

Show working

∠ACB = 180° − 47° − 63° = 70° (angles in a triangle).
∠CBD = 180° − 63° = 117° (angles on a straight line).

Triangle angle sum is used here as a supporting skill; the core Chapter 10 emphasis remains angle properties.

Classroom self-check

Can students identify the correct angle rule?

Select one answer per question, then check. Each question gives an explanation.

1. In ∠PQR, which letter represents the vertex?

2. A, O and C lie on a straight line. ∠AOB = 73°. Find ∠BOC.

3. Lines AOD and BOC intersect at O. ∠AOB = 124°. Find ∠COD.

4. Two parallel lines are cut by a transversal. One corresponding angle measures 59°. Find the other.

5. Two interior angles on the same side of a transversal between parallel lines are 116° and ∠XYZ. Find ∠XYZ.

Suggested classroom flow

Explain: Angle notation → types and measurement → intersecting lines → parallel lines → multi-step reasoning.

Demonstrate: Use the reveal-working cards, asking students to name the angle property before showing the calculation.

Practise: Use PDF 1 for textbook-style explanations, worked examples and guided practice. Use PDF 2 for additional geometry challenges.

PDF 1 — Basic Geometry Practice Workbook

13-page textbook-style companion with teaching notes, worked examples, questions and answers.

Open PDF 1 — Practice Workbook

PDF 2 — Supplementary Practice

12 additional geometry questions, four questions per page.

Open PDF 2 — Supplementary Questions
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