Congruency: Congruent Triangles
ICSE Class 7 · Mathematics · Chapter 19
What this chapter covers
- Congruent figures
- Congruence by superimposition
- Congruent line segments, angles and circles
- Congruent triangles and corresponding parts
- Criteria of congruence: SSS and SAS
- Criteria of congruence: ASA and RHS
- Using the criteria correctly
Key points
- Congruent figures have exactly the same shape and the same size; the symbol is cong.
- Moving, turning or flipping a figure does not change its congruence.
- Superimposition tests congruence by placing a traced copy of one figure on the other.
- Line segments are congruent when their lengths are equal; angles when their measures are equal; circles when their radii are equal.
- Two triangles are congruent when all three sides and all three angles match in pairs.
- In △ ABC cong △ PQR, the order of letters shows the matching: A leftrightarrow P, B leftrightarrow Q, C leftrightarrow R.
- Corresponding parts of congruent triangles are equal (CPCT).
- SSS: three sides equal in pairs.
- SAS: two sides and the included angle equal.
- ASA: two angles and the included side equal.
- RHS: in right-angled triangles, the hypotenuse and one side equal.
- AAA and SSA are not criteria of congruence.
Questions answered
What two things must be the same for two figures to be congruent?
Their shape and their size.
A square of side 4 cm and a rectangle 8 cm by 2 cm have the same area. Are they congruent? Give a reason.
No. They have the same area but different shapes, so one cannot cover the other exactly.
When are two circles congruent?
When they have the same radius (or, equally, the same diameter).
∠ P = 35° is drawn with arms 3 cm long, and ∠ Q = 35° is drawn with arms 8 cm long. Are the angles congruent?
Yes. Congruent angles need only equal measures; the lengths of the arms do not matter.
If △ ABC cong △ DEF, which side corresponds to CA and which angle corresponds to ∠ B?
CA corresponds to FD, and ∠ B corresponds to ∠ E.
In △ ABC and △ PQR, AB = PQ, BC = QR and ∠ B = ∠ Q. Which criterion shows that they are congruent?
SAS, because ∠ B and ∠ Q are the angles included between the equal sides.
In △ LMN and △ XYZ, ∠ L = ∠ X, ∠ M = ∠ Y and LM = XY. Which criterion applies, and what is ∠ Z if ∠ L = 45° and ∠ M = 85°?
ASA. ∠ Z = ∠ N = 180° - 45° - 85° = 50°.
Two right-angled triangles have hypotenuses of 10 cm and one side of 6 cm each. Are they congruent? Name the criterion.
Yes, by RHS.