Triangle congruence and geometric proof · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| congruence/ˈkɒŋɡruːəns/ | 全等 | quán děng |
Two triangular braces need to fit the same frame. Matching three angles fixes their shape but does not guarantee that their sizes agree.
- Two triangular braces need to fit the same frame. Matching three angles fixes their shape but does not guarantee that their sizes agree.
- This lesson studies congruence 全等: Equal shape and size, with matching lengths and angles.
Choose the mathematical structure
- Use SSS (three sides), SAS (two sides and their included angle), ASA (two angles and the corresponding side), or RHS (right angle, hypotenuse and one other side). Match vertices in the same order. AAA establishes similarity, not congruence. SSA generally permits more than one triangle. A proof needs a given fact, a valid criterion and a matching-part conclusion.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines congruence?
Equal shape and size, with matching lengths and angles.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For an isosceles triangle ABC with AB=AC, let D be the midpoint of BC. Triangles ABD and ACD have AB=AC, BD=DC and shared AD, so SSS gives congruence. Matching base angles ABC and BCA are therefore equal; the two angles at D are equal and form 180°, so each is 90°. To derive Pythagoras, arrange four congruent right triangles with legs a,b around a tilted square of side c inside a square of side a+b. Area gives (a+b)²=4(ab/2)+c², hence a²+b²=c². For a=3,b=4, c²=9+16=25, so c=5. These arguments establish results independently of a scale drawing.
Triangle congruence and geometric proof
Use SSS (three sides), SAS (two sides and their included angle), ASA (two angles and the corresponding side), or RHS (right angle, hypotenuse and one other side)
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
An isosceles triangle has apex angle 40°. Find either base angle.
The base angles agree, so (180-40)/2=70°.
Test a tempting shortcut
- The SAS angle must lie between the named sides. RHS uses the hypotenuse, not two arbitrary sides. A proof diagram supports the argument; it cannot establish equality just by appearance.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Two triangles with equal angles must be congruent. This claim is false. Explain which definition or assumption it violates.
Find c² for right-triangle legs 3 and 4.
The area proof gives c²=3²+4²=25.
Two triangles with equal angles must be congruent.
The SAS angle must lie between the named sides. RHS uses the hypotenuse, not two arbitrary sides. A proof diagram supports the argument; it cannot establish equality just by appearance.
Interpret a new situation
- AQA G5/G6 uses basic congruence criteria and simple geometric proofs, including isosceles base angles and Pythagoras. State every matching pair and explain which criterion applies.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find each angle at the foot D in the isosceles proof.
Congruence makes them equal; a straight line sums to 180°.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Equal shape and size, with matching lengths and angles. Choose the relationship, show the method, check its assumptions and interpret the result.