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Chapter 7: Triangles

This chapter is about congruence , when two triangles are exactly the same in size and shape, even if they sit in different places on the page. Congruence is one of the cleanest equivalence relations in geometry, and once you can establish it, a flood of useful conclusions follows: corresponding sides match, corresponding angles match, and entire proofs collapse to one or two lines.

We will meet the four standard congruence criteria:

  • SAS (Side–Angle–Side)
  • ASA (Angle–Side–Angle) and its cousin AAS
  • SSS (Side–Side–Side)
  • RHS (Right angle–Hypotenuse–Side), for right triangles

These criteria are not all equivalent; each captures a different minimum amount of information sufficient to pin down a triangle up to position. You will learn when to use which, and how to write congruence statements correctly (the order of vertices in △ABC≅△DEF\triangle ABC \cong \triangle DEF matters , it tells you which vertex corresponds to which).

Beyond congruence, the chapter studies the isosceles triangle , a triangle with two equal sides , and proves the beautiful theorem that its base angles are equal (and conversely). It also introduces inequalities in a triangle: the side opposite a larger angle is longer, and the sum of any two sides exceeds the third. These inequalities are easy to state but extremely useful in problem-solving.

By the end you should be able to look at a triangle figure and instantly spot the congruence criterion you need to invoke. You should also be able to write a short, clean congruence proof in a standard "Given / To prove / Proof" format.

What's inside

  • Congruence of triangles , definition and SAS, ASA, AAS criteria.
  • SSS criterion , three sides determine the triangle.
  • RHS criterion , for right triangles, hypotenuse and one leg are enough.
  • Isosceles triangle theorem , equal sides give equal base angles, and its converse.
  • Inequalities in a triangle , side-angle correspondence and the triangle inequality.

Key results / Formula card

ResultStatement
Congruent triangles△ABC≅△DEF\triangle ABC \cong \triangle DEF means AB=DEAB = DE, BC=EFBC = EF, CA=FDCA = FD, and all corresponding angles are equal.
SAS criterionTwo sides and the included angle of one triangle equal those of another ⇒\Rightarrow congruent.
ASA criterionTwo angles and the included side equal ⇒\Rightarrow congruent.
AAS criterionTwo angles and a non-included side equal ⇒\Rightarrow congruent.
SSS criterionAll three sides equal ⇒\Rightarrow congruent.
RHS criterionRight angle, hypotenuse, and one leg equal ⇒\Rightarrow congruent (for right triangles).
Isosceles theoremIf AB=ACAB = AC in △ABC\triangle ABC, then ∠B=∠C\angle B = \angle C.
ConverseIf ∠B=∠C\angle B = \angle C, then AB=ACAB = AC.
Side-angle correspondenceLarger angle is opposite the longer side, and vice versa.
Triangle inequalityThe sum of any two sides exceeds the third side.

Memorise this card. Nearly every problem in the chapter is one of these criteria applied to a specific figure.

Sub-topics

5 pages
01
Congruence of Triangles and the SAS Criterion
Two triangles are congruent if they coincide exactly. Learn how to write a congruence statement, and use the SAS criterion as your first proof tool.
02
The ASA and AAS Congruence Criteria
Two angles and any one side determine a triangle. Learn ASA (included side) and AAS (non-included side) and use them in proofs.
03
SSS and RHS Congruence Criteria
Three sides fix a triangle (SSS); and for right triangles, the hypotenuse and one leg are enough (RHS). Two more powerful tools for congruence proofs.
04
The Isosceles Triangle Theorem
Equal sides give equal base angles, and conversely. Two of the most elegant proofs in elementary geometry, both consequences of triangle congruence.
05
Inequalities in a Triangle
Larger angle opposite longer side; sum of any two sides exceeds the third. Two simple inequalities that govern the shape of every triangle.

Practice quiz

Answer the questions; explanations appear after each.

Quiz
Chapter 7 : Mixed practice
10 questions · pick the best answer
Q1

Two triangles are congruent if:

Q2

Which is NOT a congruence criterion?

Q3

In △ABC\triangle ABC and △DEF\triangle DEF, AB=DE,BC=EF,∠B=∠EAB = DE, BC = EF, \angle B = \angle E. The criterion is:

Q4

In △ABC\triangle ABC, AB=ACAB = AC and ∠A=50∘\angle A = 50^\circ. ∠B\angle B is:

Q5

A triangle with sides 4,9,x4, 9, x. Valid range of xx:

Q6

RHS criterion applies only to:

Q7

If △ABC≅△DEF\triangle ABC \cong \triangle DEF, which is equal to BCBC?

Q8

In a triangle with sides 5,7,95, 7, 9, the largest angle is opposite:

Q9

An equilateral triangle has each angle:

Q10

Two right triangles have hypotenuse 1313 and one leg 55. By RHS, the third side is: