Angles, congruence and similarity
Use theorem conditions to justify angle relationships and distinguish congruent from merely similar triangles.
What you will tackle
- Use vertical, supplementary and parallel-line transversal angle relationships
- Use triangle angle sums and exterior angles
- Recognise triangle congruence criteria and similarity criteria
- Connect corresponding lengths, area ratios and proof statements
Name the reason for an angle equation
Vertical opposite angles are equal, adjacent angles on a straight line sum to 180°, and triangle angles sum to 180°. Equal corresponding or alternate interior angles from a transversal require parallel lines. The exterior angle of a triangle equals the sum of its two remote interior angles.
Similarity and congruence have different conclusions
AA establishes triangle similarity; SSS in proportion and SAS in proportion with equal included angle also establish similarity. Congruence can follow SSS, SAS, ASA or AAS; for right triangles, hypotenuse-leg information can suffice. AAA gives shape, not equal size. Similarity preserves angles and scales corresponding lengths by k, areas by k².
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Two triangle angles are 48° and 67°. Find the third angle and the exterior angle at that vertex. If a parallel-line diagram has one acute angle of 48°, give the corresponding acute and supplementary obtuse angles.
Show a starting hint
State where parallelism is needed.
Compare with the explained answer
- The third triangle angle is 180 − 48 − 67 = 65°.
- The exterior angle there is 180 − 65 = 115° = 48 + 67.
- Parallel-line corresponding acute angles are 48°; supplementary obtuse angles are 132°.
Answer / response criteria: 65°; 115°; 48° and 132°.
Independent practice
Independent 1
Triangles have side sets 3,4,5 and 6,8,10. Explain similarity and its scale factor. Does AAA alone establish congruence? Would two sides and their included angle equal to another triangle establish congruence?
Compare with the explained answer
- All corresponding side ratios equal 2, so SSS similarity applies with length factor 2.
- AAA does not determine size and therefore does not establish congruence.
- Equal corresponding sides and their equal included angle give SAS congruence.
Answer / response criteria: Similar with scale factor 2; AAA is insufficient for congruence; SAS is sufficient.
Application practice
Application 1
A 1.5 m pole casts a 2 m shadow. A building casts a 14 m shadow under the same sun angle on level ground. Explain the similarity, find the building height and state one assumption.
Compare with the explained answer
- Each object and its shadow forms a right triangle, with the same sun angle.
- AA gives similar triangles, so H/14 = 1.5/2.
- H = 10.5 m, assuming vertical objects, level ground and comparable measurement conditions.
Answer / response criteria: 10.5 m under the stated similarity assumptions.
What can you explain now?
A 1.5 m pole casts a 2 m shadow. A building casts a 14 m shadow under the same sun angle on level ground. Explain the similarity, find the building height and state one assumption.
Compare with the explanation
10.5 m under the stated similarity assumptions.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide