Congruency in isosceles and equilateral triangles

key notes :

Isosceles Triangle Congruency

An isosceles triangle has two equal sides and two equal angles.

Base Angles Theorem: The angles opposite the equal sides are always equal.

Converse of Base Angles Theorem: If two angles in a triangle are equal, then the sides opposite to them are also equal.

Congruency Criteria:

  • If two isosceles triangles have equal base angles and one pair of equal sides, they are congruent by ASA (Angle-Side-Angle).
  • If they have two equal sides and the included angle, they are congruent by SAS (Side-Angle-Side).
  • If they have all three equal sides, they are congruent by SSS (Side-Side-Side).

Equilateral Triangle Congruency

  • An equilateral triangle has all three sides and all three angles equal (each angle is 60°).
  • Any two equilateral triangles with the same side length are always congruent by SSS.
  • Equilateral triangles are also isosceles, meaning they follow the base angles theorem.

Applications of Congruency in Triangles

  • Helps in proving geometric properties.
  • Used in real-life structures, bridges, and designs where symmetrical balance is required.
  • Commonly tested in geometry proofs.

Important Theorems and Properties

  • HL (Hypotenuse-Leg) Theorem applies to right-angled isosceles triangles.
  • Perpendicular bisector of an isosceles triangle passes through the opposite vertex and bisects the base.
  • Equilateral triangles have equal medians, altitudes, and angle bisectors.

Learn with an example

What is the value of p?

p =

Look at the diagram.

ST and SU are marked with one hatch mark each. So, they are congruent. By the Isosceles Triangle Theorem, the angles opposite ST and SU must also be congruent.

The angle opposite ST is ∠U and the angle opposite SU is ∠T. So, ∠U and ∠T have the same measure. From the diagram you can see that ∠T=p, so ∠U=p as well.

Now, set the sum of the interior angle measures of △STU equal to 180° and solve for p.

∠S+∠T+∠U = 180°
48°+p+p = 180° —-> Plug in ∠S=48°, ∠T=p and ∠U=p
2p+48° = 180° —–> Combine like terms
2p = 132° ——>Subtract 48° from both sides
p= 66° ——->Divide both sides by 2

So, p=66°.

What is the value of a?

a= ______°

Look at the diagram.

WX and XY are marked with one hatch mark each. So, they are congruent.

By the Isosceles Triangle Theorem, the angles oppositeWX and XY must also be congruent.

The angle opposite WX is ∠Y and the angle opposite XY is ∠W. So, ∠Y and ∠W have the same measure. From the diagram you can see that ∠Y=a, so ∠W=a as well.

Now, set the sum of the interior angle measures of △WXY equal to 180° and solve for a.

∠W+∠X+∠Y = 180°

a+46°+a = 180° Plug in ∠W=a, ∠X=46° and ∠Y=a

2a+46° = 180° Combine like terms

2a = 134° Subtract 46° from both sides

a = 67° Divide both sides by 2

So, a=67°.

What is the value of w?

w=

Look at the diagram.

∠C and ∠E are marked with one arc each. So, they are congruent.

By the Isosceles Triangle Theorem, the sides opposite ∠C and ∠E must also be congruent.

The side opposite ∠C is DE and the side opposite ∠E is CD . So, DE and CD have the same length. From the diagram you can see that the length of CD is 27 and the length of DE is w.

For these to be the same, w must equal 27.

Let’s try some problems!✍️