This chapter focuses on the congruence of triangles and properties of specific types of triangles.
- Congruence: Two figures are congruent if they have exactly the same shape and size. For triangles, congruence rules are:
- SAS (Side-Angle-Side): Two sides and the included angle.
- ASA (Angle-Side-Angle): Two angles and the included side.
- AAS (Angle-Angle-Side): Two angles and the non-included side.
- SSS (Side-Side-Side): All three sides.
- RHS (Right-Hypotenuse-Side): Right angle, hypotenuse, one side (for right triangles).
Note: AAA is NOT a congruence rule (it only ensures similarity, not congruence).
Isosceles Triangle Theorem: If two sides of a triangle are equal, the angles opposite them are equal. Conversely, if two angles are equal, the opposite sides are equal.
- Key Inequalities:
- In a triangle, the side opposite the larger angle is longer.
- The sum of any two sides is always greater than the third side (triangle inequality).
In triangles ABC and PQR, AB = PQ, BC = QR and angle B = angle Q. Prove they are congruent.
Two sides (AB = PQ, BC = QR) and the included angle (angle B = angle Q) are equal. By SAS, triangle ABC is congruent to triangle PQR.
In an isosceles triangle ABC, AB = AC. If angle B = 65 degrees, find angle A.
Since AB = AC, angle B = angle C = 65 degrees. Angle A = 180 - 65 - 65 = 50 degrees.
Can a triangle have sides 3 cm, 4 cm and 8 cm?
3 + 4 = 7 < 8. The sum of two sides is less than the third. No, this is not a valid triangle.
In triangle ABC, AB = AC and D is the midpoint of BC. Prove that AD is perpendicular to BC.
Triangles ABD and ACD: AB = AC (given), BD = CD (D is midpoint), AD is common.
By SSS, triangles are congruent, so angle ADB = angle ADC.
Since they are supplementary (form a straight line), each = 90 degrees. Hence AD is perpendicular to BC.
In triangle PQR, angle P > angle R. Which side is longer, QR or PQ?
The side opposite the larger angle is longer. Angle P > angle R means QR > PQ.
Two right triangles have equal hypotenuse and one equal leg. Are they congruent?
Yes — by RHS, two right triangles with equal hypotenuse and one equal side are congruent.
In triangle ABC, D is the midpoint of BC, and AD = BD = DC. Find angle A.
AD = BD means triangle ABD is isosceles and AD = DC means triangle ACD is isosceles. Since AD = BD = DC = (1/2)BC, and using properties, angle BAC = 90 degrees (angle in a semicircle concept).
Common mistakes
Students often apply AAA as a congruence rule — it is not. Also, when writing congruence, maintain correct correspondence: triangle ABC is congruent to triangle DEF means A-D, B-E, C-F.
Summary
Congruence rules (SAS, ASA, AAS, SSS, RHS) allow us to prove triangles identical. Key properties of isosceles triangles and inequalities in triangles are powerful tools in geometric proofs.