If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.
Proofs involving isosceles triangles often require special consideration because an isosceles triangle has several distinct properties that do not apply to normal triangles.(More about triangle types) Therefore, when you are trying to prove that two triangles are congruent, and one or both triangles, are isosceles you have a few theorems that you can use to make your life easier.
Isosceles Triangle
An isosceles triangle has two congruent sides and two congruent angles. The congruent angles are called the base angles and the other angle is known as the vertex angle. $$ \angle $$BAC and $$ \angle $$BCA are the base angles of the triangle picture on the left. The vertex angle is $$ \angle $$ABC
Isosceles Triangle Theorems
The Base Angles TheoremIf two sides of a triangle are congruent, then the angles opposite those sides are congruent.
A triangle that has two sides of the same measure and the third side with a different measure is known as an isosceles triangle. The isosceles triangle theorem in math states that in an isosceles triangle, the angles opposite to the equal sides are also equal in measurement. We will be learning about the isosceles triangle theorem and its converse in this article.
What is Isosceles Triangle Theorem?
Isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are also congruent. To understand the isosceles triangle theorem, we will be using the properties of an isosceles triangle for the proof as discussed below.
Isosceles Triangle Theorem Proof
Let's draw an isosceles triangle with two equal sides as shown in the figure below.
Given: ∆ABC is an isosceles triangle with AB = AC.
Construction: Altitude AD from vertex A to the side BC.
To Prove: ∠B = ∠C.
Proof: We know, that the altitude of an isosceles triangle from the vertex is the perpendicular bisector of the third side. Thus, we can conclude that, ∠ADB = ∠ADC = 90º ----------- (1) BD = DC ---------- (2) Consider ∆ADB and ∆ADC AB = AC [Given] AD = AD [common side] BD = DC [From equation (2)]
Thus, by SSS congruence we can say that,
∆ADB ≅ ∆ADC
By CPCT, ∠B = ∠C.
Hence, we have proved that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are equal.
Isosceles Triangle Theorem Converse
The converse of isosceles triangle theorem states that if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal. This is exactly the reverse of the theorem we discussed above. We will be using the properties of the isosceles triangle to prove the converse as discussed below.
Converse of Isosceles Triangle Theorem Proof
Let's draw a triangle with two congruent angles as shown in the figure below with the markings as indicated.
Given: ∆ABC with ∠B = ∠C.
Construction: Altitude AD from vertex A to the side BC.
To Prove: AB = AC
Proof: We know that the altitude of a triangle is always at a right angle with the side on which it is dropped. Hence, ∠ADB = ∠ADC = 90º ----------- (1) Consider ∆ADB and ∆ADC, ∠B = ∠C [Given] AD = AD [common side] ∠ADB = ∠ADC = 90º [From equation (1)]
Thus, by AAS congruence we can say that,
∆ADB ≅ ∆ADC
By CPCT, AB = AC
Hence we have proved that, if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal.
Related Articles
Check these articles related to the concept of the isosceles triangle theorem.
- Isosceles Triangle
- Perpendicular Bisector
- Congruence in Triangles
-
Example 1: In the given figure below, find the value of x using the isosceles triangle theorem.
Solution: According to the given figure,
In ∆XYZ, we see that XY = XZ = 12 cm
According to the isosceles triangle theorem, if two sides of a triangle are congruent, then the angles opposite to the congruent sides are equal.
Thus, ∠Y = ∠Z [Since XY = XZ]
∠Y = 35º, ∠Z = x
Thus, ∠Y = ∠Z = 35º.
Hence the value of x is 35º.
-
Example 2: If ∠P and ∠Q of ∆PQR are equal to 70º and QR = 7.5 cm, find the value of PR.
Solution: Let's draw a figure according to the given question,
Given that, in ∆PQR, ∠P = ∠Q = 70º.
According to the isosceles triangle theorem converse, if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal.
Thus, PR = QR [Since, ∠P = ∠Q]
But, QR = 7.5 cm
Therefore the value of PR = 7.5 cm.
View Answer >
go to slidego to slide
Want to build a strong foundation in Math?
Go beyond memorizing formulas and understand the ‘why’ behind them. Experience Cuemath and get started.
Book a Free Trial Class
FAQs on Isosceles Triangle Theorem
Isosceles triangle theorem states that, if two sides of an isosceles triangle are equal then the angles opposite to the equal sides will also have the same measure.
How to Prove Isosceles Triangle Theorem?
Isosceles triangle theorem can be proved by using the congruence properties and properties of an isosceles triangle. An isosceles triangle can be drawn, followed by constructing its altitude. The two triangles now formed with altitude as its common side can be proved congruent by SSS congruence followed by proving the angles opposite to the equal sides to be equal by CPCT.
What is the Converse of Isosceles Triangle Theorem?
The converse of isosceles triangle theorem states that, if two angles of a triangle are equal, then the sides opposite to the equal angles of a triangle are of the same measure.
How to Prove the Converse of the Isosceles Triangle Theorem?
The converse of the isosceles triangle theorem can be proved by using the congruence properties and properties of an isosceles triangle. An isosceles triangle can be drawn, followed by constructing its altitude. The two triangles now formed with altitude as its common side can be proved congruent by AAS congruence followed by proving the sides opposite to the equal angles to be equal by CPCT.
How to find Angles using Isosceles Triangle Theorem?
The angles of an isosceles triangle add up to 180º according to the angle sum property of a triangle. The angles opposite to the equal sides of an isosceles triangle are considered to be an unknown variable 'x'. Now, if the measure of the third (unequal) angle is given, then the three angles can be added to equate it to 180º to find the value of x that gives all the angles of a triangle. For example: Let the unequal angle of an isosceles triangle be 50º. The other angles can be considered as x each as they are equal. By using the angle sum property, 50º + x + x = 180° 2x = 180º - 50º 2x = 130º x = 65º
Thus, the angles of the isosceles triangle are 65º, 65º, and 50º.