Vertically Opposite Angles

Overview

In this seventh lesson of “Angles: Measure, Build, Reason”, students investigate pairs of angles formed by intersecting lines. They use folding, tracing, measurement and dynamic diagrams to discover that vertically opposite angles are equal, then apply this relationship to solve unknown-angle problems.

Learning intentions

• WALT identify vertically opposite angles formed by intersecting lines.
• WALT explain why vertically opposite angles are equal.
• WALT use angle facts on straight lines and at a point to find unknown angles.
• WALT communicate mathematical reasoning using diagrams, measurements and equations.

Success criteria

• I can find and colour-code vertically opposite angle pairs.
• I can measure angles and predict the size of their vertically opposite angles.
• I can use a known angle to calculate an unknown angle.
• I can explain my answer using “vertically opposite angles are equal”.

Curriculum links

• Geometry: identify, describe, measure and reason about angles and relationships between lines.
• Mathematical and statistical practices: represent ideas, notice patterns, reason, explain and justify conclusions.
• Number and algebra connections: use addition and subtraction to solve missing-angle problems.
• Communication and participation: discuss strategies, compare solutions and respond to others’ reasoning.

Lesson structure (45 minutes)

1. 0–5 minutes – Hook and prior knowledge

Open with the hook and learning intention slides. Display two intersecting lines with one angle marked and ask: “Without measuring, which other angle might be the same size? How do you know?” Students think silently, then share with a partner. Briefly revisit the terms line, vertex, angle, straight angle and angle at a point.

2. 5–12 minutes – Fold and trace the pattern

In groups of three or four, students draw two crossing lines on a sheet of paper. They trace or lightly shade each of the four angles, then fold the paper along one line and compare the opposite regions. Use different colours for the two pairs of opposite angles. Ask: “What moves onto what? What stays the same?” Explain that angles opposite each other, formed by two intersecting lines, are called vertically opposite angles. They are equal; they are not necessarily right angles.

3. 12–19 minutes – Measure and predict

Display the worked example in the folding, tracing and measurement slides. Students measure one angle on their drawing with a protractor and record the size of its opposite angle. They predict the size before measuring, then check. Invite groups to report evidence. Emphasise that the equal angles share only the vertex and are opposite each other, not next to each other.

4. 19–27 minutes – Explicit teaching and guided examples

Model a colour-coded diagram: if the top-left angle is 65°, the bottom-right vertically opposite angle is also 65°. Then model a straight-line example: if one angle is 115°, the adjacent angle is (180° - 115° = 65°), so its vertically opposite angle is 65°. Work through the examples in the guided-practice slides. Students show answers on mini-whiteboards and explain which fact they used: “angles on a straight line total 180°” or “vertically opposite angles are equal”.

5. 27–38 minutes – Paired problem solving

Distribute the vertically opposite angles practice worksheet. Students work in pairs, taking turns as “solver” and “checker”. The worksheet should progress from colour-matching opposite pairs, to measuring and predicting, then to missing-angle questions involving straight lines and vertically opposite angles. Pause halfway for a quick check of a common error: matching neighbouring angles instead of opposite angles. Confer with groups and ask, “What do you know first?” and “Which angle fact supports your calculation?”

6. 38–43 minutes – Challenge and discussion

Display the multi-step diagrams in the challenge and discussion slides. Students solve one example independently, then compare methods with a partner. Include a diagram with angles at a point, a straight line and a vertically opposite pair. Select two solutions to discuss, focusing on clear labelling and the order of reasoning rather than speed.

7. 43–45 minutes – Exit check and reflection

Return to the plenary slide. Students answer: “One angle is 72° in a pair of intersecting lines. What is its vertically opposite angle? Explain why.” Collect responses or scan them as students leave. Ask students to complete the sentence: “Today I know vertically opposite angles are equal because…”

Resources

• the complete lesson slide deck
• the vertically opposite angles practice worksheet
• Plain paper for folding and drawing
• Pencils, rulers and coloured pencils
• Protractors
• Mini-whiteboards and pens
• Visualiser or board
• Prepared intersecting-line diagrams for modelling

Assessment

• Observe folding, tracing and discussion: can students distinguish opposite angles from adjacent angles?
• Check worksheet reasoning, not just answers, particularly correct use of 180° on a straight line.
• Use the exit response to identify whether students can state and apply the equal-angle relationship independently.

Differentiation

• Support: provide pre-drawn intersecting lines, colour-coded matching pairs, a word bank and the sentence frame “The angle opposite ___° is ___° because…”. Work with a teacher-led group on one fact at a time.
• Support for EAL and SEN: use gestures and repeated visual examples; say and display “opposite”, “equal”, “straight line” and “at a point”. Allow tracing, enlarged diagrams and verbal explanations instead of lengthy written responses.
• Core: require students to mark the equal pair before calculating and write the fact used beside each answer.
• Extension: ask students to solve multi-step diagrams with several angles at a point, explain two possible solution paths, or create a diagram where the unknown angle is found using both a straight-line fact and vertically opposite angles.