5.1 Congruency
Congruency refers to geometric figures that have the exact same shape and size, regardless of their position, orientation, or arrangement. In daily life, real-world objects like identical floor tiles, the Petronas Twin Towers, or the front and rear tyres of a car exhibit congruent properties.
Learning Objectives
- Differentiate between congruent and non-congruent shapes based on corresponding sides and corresponding angles.
- Make and verify conjectures of triangle congruency using specific congruence rules: SSS, SAS, ASA, AAS, AAA (under equal area condition), and SSA/RHS (under equal area condition).
- Solve various geometric and real-world problems involving congruency.
Summary of Triangle Congruency Rules
A side-by-side comparison of valid congruency rules versus conditional rules.
All three pairs of corresponding sides are equal.
Two sides and the included angle are equal.
Two angles and the included side are equal.
Two angles and a non-included side are equal.
Right angle, hypotenuse and one side are equal.
Valid only if areas are equal.
Valid only if areas are equal.
Brief Explanation
What is Congruency?
Two geometric shapes are congruent when they have the exact same size and shape. If you place one directly over the other, they overlap completely.
Congruent Polygons
Two polygons are congruent if and only if all corresponding sides have the same length and all corresponding angles have the same size.
Notation
If triangle \(ABC\) is congruent to triangle \(PQR\), it is written as \(\Delta ABC \cong \Delta PQR\).
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
A triangle has sides measuring \(5\text{ cm}\), \(12\text{ cm}\), and \(13\text{ cm}\). Another triangle has sides measuring \(12\text{ cm}\), \(13\text{ cm}\), and \(5\text{ cm}\). Are these two triangles congruent, and which rule justifies it?
Why is Angle-Angle-Angle (AAA) generally insufficient to prove that two triangles are congruent?
In right-angled triangle \(ABC\) (\(\angle B = 90^\circ\)) and right-angled triangle \(DEF\) (\(\angle E = 90^\circ\)), \(AC = DF = 10\text{ cm}\) and \(AB = DE = 6\text{ cm}\). State the special congruency rule that proves \(\Delta ABC \cong \Delta DEF\).
Common Mistakes
- Confusing SAS with SSA: placing the given angle between the two known sides gives SAS, which is valid. Placing the angle outside the two sides gives SSA, which does not guarantee congruency unless areas are identical.
- Assuming AAA implies congruency: if all three angles match, the triangles are merely similar unless a side length or area is proven equal.
- Mismatched corresponding vertices: writing \(\Delta ABC \cong \Delta DEF\) without ensuring vertex \(A\) matches \(D\), \(B\) matches \(E\), and \(C\) matches \(F\).
Misconceptions
Two shapes with the same area are always congruent.
Having the same area does not mean shapes are congruent. A rectangle of \(4\text{ cm} \times 9\text{ cm}\) and a rectangle of \(6\text{ cm} \times 6\text{ cm}\) both have an area of \(36\text{ cm}^2\), but their side dimensions and shapes differ completely.
If two shapes are flipped or rotated, they are no longer congruent.
Congruence depends strictly on shape and size. Position, orientation, translation, rotation, and reflection do not affect whether two shapes are congruent.
Summary
- Congruence: Polygons are congruent if all corresponding side lengths and angle sizes are identical (\(\cong\)).
- SSS: 3 pairs of equal sides.
- SAS: 2 pairs of equal sides and the included equal angle.
- ASA: 2 pairs of equal angles and the included equal side.
- AAS: 2 pairs of equal angles and 1 non-included equal side.
- RHS: Right-angle, equal hypotenuse, and 1 equal corresponding side.
- Conditional Rules: AAA and SSA can only determine congruency if the areas of the triangles are proven equal.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following conditions is always valid to determine that two triangles are congruent without needing their areas to be given?
2 Given \(\Delta ABC \cong \Delta PQR\), if \(AB = 7\text{ cm}\), \(BC = 9\text{ cm}\), and \(\angle ABC = 45^\circ\), which statement must be true?
B. Fill in the Blanks
3 The mathematical symbol used to denote that two triangles are congruent is .
Answer for question 34 The special congruence rule for right-angled triangles where the hypotenuse and one corresponding non-hypotenuse side are equal is known as .
Answer for question 4C. Drag and Drop: Match the Rule to Its Description
Drag each congruency rule to the correct description. On a phone, tap the answer choice first, then tap the description box.
