Year 10 - Geometric reasoning and proofs - Free practice

Year 10 Geometric Reasoning and Proofs Worksheets & Practice

Build logical arguments about shapes, step by step, with worked solutions every time.

Quick answer

How do you write geometric proofs involving triangles and quadrilaterals?

State a known fact (given information or a theorem), then chain logical steps until you reach the conclusion. For triangles, use SSS, SAS, AAS or RHS to prove congruence, or AA, SAS or SSS to prove similarity. For quadrilaterals, use properties such as opposite sides equal or parallel to prove a shape is a parallelogram or rhombus.

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Worked examples

Year 10 geometric reasoning and proofs: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Congruence proof using SAS
Triangle $ABC$ has $AB = 5$ cm, $BC = 7$ cm and $\angle ABC = 48°$ . Triangle $DEF$ has $DE = 5$ cm, $EF = 7$ cm and $\angle DEF = 48°$ . Prove that $\triangle ABC \cong \triangle DEF$ .
Show worked solution
Key idea: Two triangles are congruent by SAS if two pairs of corresponding sides and the included angle are equal.
1. $AB = DE = 5$ cm (given)
2. $BC = EF = 7$ cm (given)
3. $\angle ABC = \angle DEF = 48°$ (given), and this angle is included between the two known sides in each triangle.
4. Therefore $\triangle ABC \cong \triangle DEF$ (SAS).
Answer: $\triangle ABC \cong \triangle DEF$ by SAS.
Example 2 · Congruence proof using ASA
In triangles $PQR$ and $XYZ$ , $\angle QPR = \angle YXZ$ , $PQ = XY$ and $\angle PQR = \angle XYZ$ . Prove that $\triangle PQR \cong \triangle XYZ$ .
Show worked solution
Key idea: Two triangles are congruent by ASA if two pairs of corresponding angles and the included side are equal.
1. $\angle QPR = \angle YXZ$ (given).
2. $\angle PQR = \angle XYZ$ (given).
3. $PQ = XY$ (given), and this side lies between $\angle QPR$ and $\angle PQR$ in $\triangle PQR$ , and between $\angle YXZ$ and $\angle XYZ$ in $\triangle XYZ$ , so it is the included side in each triangle.
4. Therefore $\triangle PQR \cong \triangle XYZ$ (ASA).
Answer: $\triangle PQR \cong \triangle XYZ$ by ASA.
Example 3 · Proving a property of a parallelogram
Quadrilateral $ABCD$ has two pairs of parallel sides: $AB \parallel DC$ and $AD \parallel BC$ . The diagonal $AC$ is drawn, forming triangles $ABC$ and $CDA$ . Prove that $AB = DC$ .
Show worked solution
Key idea: Use alternate angle theorems with the diagonal as a transversal to establish congruent triangles, then conclude equal sides.
1. Since $AB \parallel DC$ and $AC$ is a transversal, $\angle BAC = \angle DCA$ (alternate angles).
2. Since $AD \parallel BC$ and $AC$ is a transversal, $\angle BCA = \angle DAC$ (alternate angles).
3. $AC = CA$ (common side), and this side is included between $\angle BAC$ and $\angle BCA$ in $\triangle ABC$ , and between $\angle DCA$ and $\angle DAC$ in $\triangle CDA$ .
4. Therefore $\triangle ABC \cong \triangle CDA$ (ASA).
5. Hence $AB = DC$ (corresponding sides of congruent triangles).
Answer: $AB = DC$ , proven via $\triangle ABC \cong \triangle CDA$ by ASA.

Common mistakes

Where students usually lose marks

Confusing ASA and AAS

Two angles and a side are given, so I will use ASA.

ASA requires the known side to be between the two known angles (the included side). AAS requires the known side to be not between the two known angles. Always check exactly where the side sits relative to the two angles before choosing the condition.

Listing facts without linking them to the condition

$AB = DE$ , $\angle A = \angle D$ , $\angle B = \angle E$ , therefore congruent.

A valid proof must name the congruence condition (SSS, SAS, ASA, AAS or RHS) and match each fact to a specific part of that condition. State which angle is included, or which side is not included, so the reader can see why the condition applies.

Using SSS when a side is not actually given

Both triangles share side $AC$ , so I know three sides and can use SSS.

Sharing a common side gives you only one pair of equal sides. You need all three pairs of sides to be equal for SSS. If you only have one or two pairs of sides, look for angle information and consider SAS, ASA or AAS instead.

Assuming rather than proving parallel lines give equal angles

$\angle BAC = \angle DCA$ because the lines look parallel.

You must state the theorem explicitly: for example, 'alternate angles, $AB \parallel DC$ , transversal $AC$ '. Parallel lines must be given or already proven before you can apply co-interior, alternate or corresponding angle theorems.

Try one yourself

Your turn

In $\triangle ABC$ , $AB = AC$ and $M$ is the midpoint of $BC$ . It can be proved that $\triangle ABM \equiv \triangle ACM$ by SSS. If $AB = 10$ cm and $BC = 12$ cm, what is the length of $AM$ in centimetres, correct to 2 decimal places?

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About this worksheet

Why we built it

Geometric reasoning and proofs ask Year 10 students to move beyond finding answers and start explaining why those answers must be true. These worksheets guide students through deductive reasoning with triangles, quadrilaterals and parallel lines, building the careful, structured arguments required by the curriculum. Every question comes with a full worked solution so students can see exactly how each logical step connects to the next.

What's covered

Sub-skills your student will practise

✓Congruence proofs using the four triangle tests (SSS, SAS, AAS, RHS)
✓Similarity proofs and identifying the correct similarity condition
✓Proving properties of triangles using angle and side theorems
✓Proving properties of quadrilaterals such as parallelograms and rhombuses
✓Applying parallel line theorems (co-interior, alternate, corresponding) in proofs
✓Using deductive reasoning to solve multi-step spatial problems

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Frequently asked

Questions parents ask about geometric reasoning and proofs

What is deductive reasoning in Year 10 geometry?

Deductive reasoning means building a logical argument from known facts and theorems to reach a conclusion that must be true. In Year 10 geometry, students chain together angle rules, congruence tests and side properties in a structured proof rather than just stating an answer.

What are the four triangle congruence tests students need to know?

The four tests are SSS (three sides equal), SAS (two sides and the included angle equal), AAS (two angles and a corresponding side equal) and RHS (right angle, hypotenuse and one other side equal). Students need to state which test they are using and show that all required conditions are met.

How is a geometry proof different from just solving for a missing angle?

Solving for an angle gives a numerical answer, while a proof shows that a general statement is always true. A proof lists each step with a reason, such as a theorem or given information, so the argument holds for any figure that satisfies the starting conditions.

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