Topics
7. Triangles
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United States of America
Grades 9-12

7.02 Isosceles and equilateral triangles

Lesson
Practice

Introduction

We will explore the special properties of isosceles and equilateral triangles and prove them to be true. Once we have convinced ourselves of some of these special properties, we will use them and related properties to solve mathematical and contextual problems.

Isosceles and equilateral triangles

Following are the properties and theorems relating to isosceles and equilateral triangles.

Isosceles triangle

A triangle containing at least two equal-length sides

A triangle with two congruent sides. The congruent sides are highlighted.
Equilateral triangle

A triangle with three equal-length sides. Equilateral triangles are a sub-class of isosceles triangles

A triangle with three congruent sides.
Legs of an isosceles triangle

The sides of an isosceles triangle that are equal in length

A triangle with two congruent sides. The congruent sides are highlighted.
Base angles of an isosceles triangle

The angles that are opposite the legs of an isosceles triangle

A triangle with two congruent sides. The angles opposite the congruent sides are highlighted.

Exploration

Drag the dots, then move the orange slider to change the angle size and move the blue slider to build a triangle.

Loading interactive...
  1. What is true about the orange angles?
  2. What else do you notice about this triangle?

We can explore constructing triangles with two congruent sides or with two congruent angles, and notice the relationship between the base angles and congruent sides of an isosceles triangle. This relationship is summarized in the following theorems:

Base angles theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent

Triangle A B C with congruent sides A B and A C.

In this triangle, since \overline{AB} and \overline{AC} are legs of the triangle, the theorem tells us that the base angles \angle ABC and \angle ACB are congruent.

Converse of base angles theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent

Triangle A B C. Angles B and C are highlighted.

In this triangle, since \angle ABC and \angle ACB are congruent, the theorem tells us that \overline{AB} \cong \overline{AC}.

While we can understand these two theorems are valid using constructions or dynamic geometric software, we will also formally prove these theorems in a later lesson.

We can apply these two theorems repeatedly to show the corollaries that follow about equilateral triangles. A corollary is a proposition inferred immediately from something that has already been proven.

Corollary to the base angles theorem

If a triangle is equilateral, then it is equiangular

Corollary to the converse of the base angles theorem

If a triangle is equiangular, then it is equilateral

A triangle with 3 congruent sides and angles.

Examples

Example 1

Prove the corollary to the base angles theorem:

If ABC is an equilateral triangle, prove all angles are congruent and equal to 60 \degree.

Worked Solution
Create a strategy

Draw a diagram with the characteristics given.

Triangle A B C with all sides congruent.

First, we need to prove that all angles are congruent. Then we can determine the measure of each angle.

Apply the idea
To prove: All angles of an equilateral triangle are congruent
StatementsReasons
1.\overline{AB} \cong \overline{BC}Given
2.\triangle{ABC} is isoscelesDefinition of isosceles triangles
3.\angle A \cong \angle CBase angles theorem
4.\overline{BC} \cong \overline{AC}Given
5.\triangle{ABC} is isoscelesDefinition of isosceles triangles
6.\angle A \cong \angle BBase angles theorem
7.\angle A \cong \angle B \cong \angle CTransitive property of congruence

Since all three angles of an equilateral triangle are congruent, it follows from the triangle sum theorem that their sum must be 180 \degree. Therefore, the measure of each angle in an equilateral triangle is {\dfrac{180}{3}}^{\degree} = 60 \degree.

Example 2

Is \triangle{CDE} isosceles, equilateral, or neither? Justify your answer in a proof.

Horizontal segment A G with points C and E on A G. Point B above segment A G and on the right of A is drawn. Ray B A is drawn. Point D below A G and on the right of B is drawn. Segment B D intersects A G at C. A point F directly above G is drawn. Ray D F is drawn intersecting A G at E. Angle E F G has a measure or 45 degrees. F G and E G are congruent. Angles C A B, A B C, and B C A are congruent.
Worked Solution
Create a strategy

To prove \triangle{CDE} isosceles or equilateral, we need to show congruent angles or congruent sides.

Since we have some angle measures in the diagram, let's try to find the angle measures in \triangle{CDE}.

Apply the idea
Two flow chart proofs. Left flow chart proof with 6 levels and the reason below each step. Right flow chart proof with 6 levels and the reason below each step. Speak to your teacher for more details.

Example 3

Consider the triangles in the diagram shown and given \overline{QP} \cong \overline{QT}:

Line P S with point T on P S. A point Q is above segment P T. Triangle P Q T is drawn with angle Q T P measuring 60 degrees. A point R is drawn on segment Q T. Triangle T R S is drawn. Segments R T and T S are congruent as well as angles T R S and R S T.
a

Identify all possible pairs of congruent angles.

Worked Solution
Create a strategy

Look at angles already marked congruent and use information from the triangles to determine other congruent pairs of angles.

Apply the idea

\angle TRS \cong \angle RST is given from the diagram.

Since \overline{QP} \cong \overline{QT} (given), we know that \triangle QPT is isosceles, so we have\angle QPT \cong \angle QTP by the base angles theorem and therefore m\angle QPT = m\angle QTP = 60 \degree. This would also give us:

\displaystyle m\angle PQT \displaystyle =\displaystyle 180 \degree - \angle QPT - \angle QTPAngle sum of \triangle QTP
\displaystyle =\displaystyle 180 \degree - 60\degree - 60\degreeSubstitute m\angle QPT = m\angle QTP = 60 \degree
\displaystyle =\displaystyle 60\degreeEvaluate the subtraction

We have shown \triangle QTP is equiangular. So, we can also state that \angle QTP \cong \angle TPQ, \angle QTP \cong \angle PQT, and \angle TPQ \cong \angle PQT.

b

If m \angle QTS = (7x+85) \degree, find the value of x.

Worked Solution
Create a strategy

\angle QTP and \angle RTS are a linear pair because they lie on \overleftrightarrow{PS}, we know their sum is 180 \degree. We can use this to solve for m \angle RTS and find the value of x.

Apply the idea
\displaystyle m \angle QTP + m \angle RTS\displaystyle =\displaystyle 180\angle QTP and \angle RTS are a linear pair
\displaystyle 60 + m \angle RTS\displaystyle =\displaystyle 180Substitution
\displaystyle m \angle RTS\displaystyle =\displaystyle 120Subtract 60 from both sides
\displaystyle 7x+85\displaystyle =\displaystyle 120Transitive property of equality
\displaystyle 7x\displaystyle =\displaystyle 35Subtract 85 from both sides
\displaystyle x\displaystyle =\displaystyle 5Divide both sides by 7

Example 4

Consider the polygon shown:

Triangles A B C and A C D, sharing a common side A C. A B has a length of 12.5. Angle A D C has a measure of 72 degrees. A D, A C, A B, and B C are congruent.
a

If m \angle ACD = (2x+56) \degree, find the value of x.

Worked Solution
Create a strategy

Since \triangle ACD is an isosceles triangle, \angle ADC \cong \angle ACD by the base angles theorem and therefore m \angle ADC = m \angle ACD by definition of congruency. Use the fact that m \angle ACD = 72 \degree to solve for x.

Apply the idea
\displaystyle m \angle ACD\displaystyle =\displaystyle 72\triangle ACD is isosceles
\displaystyle 2x + 56\displaystyle =\displaystyle 72Transitive property of equality
\displaystyle 2x\displaystyle =\displaystyle 16Subtract 56 from both sides
\displaystyle x\displaystyle =\displaystyle 16Divide by 2 on both sides
b

If AD=3y-9.25, find the value of y.

Worked Solution
Create a strategy

Based on the diagram, \overline{AB} \cong \overline{AD} and we are given that AB=12.5, so write and solve an equation using AD and AB.

Apply the idea
\displaystyle AB\displaystyle =\displaystyle ADDefinition of congruence
\displaystyle 12.5\displaystyle =\displaystyle 3y-9.25Substitution
\displaystyle 21.75\displaystyle =\displaystyle 3yAdd 9.25 to both sides
\displaystyle 7.25\displaystyle =\displaystyle yDivide both sides by 3
Idea summary

We can use the following theorems to solve problems:

  • The base angles theorem states that if two sides of a triangle are congruent, then the angles opposite them are congruent
  • The converse of base angles theorem states that if two angles of a triangle are congruent, then the sides opposite them are congruent
  • An equilateral triangle is a sub-class of isosceles triangles that has three equal-length sides and three 60\degree interior angles. It may also be referred to as an equiangular triangle

Outcomes

G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

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