Congruent Triangles and Their Properties: Quick Revision

A brief note on Congruent Triangles and Their Properties. Read it before a test. Try each question at the end before you open its answer.

Congruence

• Congruent triangles fit exactly on each other. In $\triangle ABC \cong \triangle PQR$, the order of the letters shows which corners match.
• CPCT: corresponding parts of congruent triangles are equal.
• SAS: two sides and the angle between them. ASA: two angles and the side between them. AAS: two angles and a side elsewhere.
• SSS: three sides. RHS: a right angle, the hypotenuse and one side.
• Not rules: AAA (the same angles, any size) and SSA (two sides and an angle that is not between them).

Isosceles triangles

• Angles opposite equal sides are equal, and sides opposite equal angles are equal.
• The perpendicular from the apex bisects the base and the apex angle.
• An equilateral triangle has three angles of $60^\circ$.

Sides and angles

• The longer side is opposite the larger angle, and the larger angle is opposite the longer side.
• Any two sides together are longer than the third. The third side lies between $|a - b|$ and $a + b$.

A worked example

In triangle $ABC$, $AB = AC$ and $D$ is the middle point of $BC$. Show that $\angle BAD = \angle CAD$.

1. $AB = AC$ and $BD = DC$. These are given.
2. $AD = AD$. This is the common side.
3. So $\triangle ABD \cong \triangle ACD$, by SSS.
4. Therefore $\angle BAD = \angle CAD$, by CPCT.

Practice with new numbers

A program to try

The program compares the sides of two triangles given by their corners. Equal sides mean the triangles are congruent by SSS.

Mistakes to avoid

• Writing the letters in the wrong order when you state the congruence.
• Using SSA or AAA as if they were rules.
• Skipping the common side as a step of the proof.
• Allowing a flat triangle: the longest side must be less than the other two together.

Check yourself

1. In $\triangle ABC \cong \triangle PQR$, which side matches $BC$?

2. Triangle $ABC$ has $AB = AC$ and $\angle A = 30^\circ$. Find $\angle B$.

3. Can the lengths $4$, $6$ and $11$ form a triangle?

4. Two triangles have equal angles of $50^\circ$, $60^\circ$ and $70^\circ$. Are they congruent?