Proof of Isosceles Triangle Theorem with Solved Examples
An isosceles triangle is one of the many varieties of triangle differentiated by the length of their sides. Today we will learn more about the isosceles triangle and its theorem. An isosceles triangle is known for its two equal sides. The peak or the apex of the triangle can point in any direction. Check this example:
Here we have an isosceles triangle ABC, where side AB is congruent to side BC, which is the main reason the triangle is isosceles. These are called legs; if they are equal, it is an isosceles triangle.
Hence, length of AB = length of BC
Some of the Basics About the Isosceles Triangle are as Follows:
The triangle has two equal sides, with the base as the third, unequal side.
The angles opposite the two equal sides will always match
An isosceles triangle is called a right isosceles triangle when its third angle is 90 degrees.
Properties of an Isosceles Triangle
Here we have an isosceles triangle ABC to explore the parts. Let us discuss some of the properties of an isosceles triangle.
Like any triangle, ABC has three interior angles those are ∠A, ∠B, and ∠C
All three interior angles are acute angles.
It also has three sides, and those are AB, BC, and AC
Side AB is congruent to side BC; therefore, we refer to those twins as legs.
The third side AC is known as the base, even if the triangle is not sitting on that side.
The angles formed between the base and leg ∠A and ∠C are called base angles
Isosceles Triangle Theorem and Its Proof
Theorem 1 - “Angle opposite to the two equal sides of an isosceles triangle are also equal.”
Proof: consider an isosceles triangle ABC, where AC=BC. we will have to prove that angles opposite to the sides AC and BC are equal, i.e., ∠CAB = ∠CBA
To test this mathematically, we will have to introduce a median line. Here, a line must be constructed from an interior angle to the midpoint of the opposite side, which is the base side AB.
We will find Point M on the base side AB, where we can construct the line segment CM.
After constructing, we get two triangles; those are CAM and CBM.
Now,
CA = CB (Given)
AM= BM (median)
CM = CM (reflective property)
Thus, triangle CAM = triangle CBM.
SO, ∠CAM = ∠CBM (CPCTC)
Putting this in words we have shown that the three sides of triangle CAM are congruent to triangle CBM, which means here you have the SIDE SIDE SIDE Postulate, which gives the congruence. So, if two triangles are congruent, then the parts corresponding to the congruent triangle are congruent (CPCTC). This gives you A is congruent to ∠B.
Theorem 2
It is the converse of the isosceles triangle theorem
“Sides opposite to the two equal angles of a triangle are equal.”
Proof: Consider an isosceles triangle DEF, here we must prove that side DE = side DF and DEF is isosceles.
The converse of the isosceles triangle theorem says that if two angles of the triangle are equal, then the opposite side is the same.
Now you are wondering whether this statement is true or not. Not every converse statement of an original statement is true. The converse could be either true or false if the original statement is false.
To prove the converse statement lets construct a bisector DG which meets the side EF at right angles.
∠DEF = ∠DFE (given)
DG = DG (reflective property)
∠DGE = ∠DGF = 90 degrees (By Construction)
Thus, triangle DEG = triangle DFG (By ASA congruency)
So, DE = DF (By CPCTC)
In simpler words, we have two small right-angle triangles, DEG and DFG, instead of one big isosceles triangle. Since the line DG is an angle bisector, it makes <DGE and ∠DGF congruent to each other. Since the line segment, DG, is used in both right-angle triangles, it is congruent to itself. This would be angle, side, angle (ASA) theorem. With the triangles being congruent, their corresponding part also becomes congruent, which makes DE=DF. It is worth noting that isosceles theorem converse is true.
Solved Isosceles Triangle Questions
In triangle PQR, we have PQ=PR and ∠PQR =47degrees. Find ∠QPR
Solution: By the isosceles triangle theorem, we have 47 degrees =PQR =PRQ. As the angles in a triangle sum up to 180 degrees, we have
∠QPR = 180 degree – ( ∠PQR + ∠PRQ)
=180 – 2 x 47 = 86 degree
Fun facts About Isosceles Triangle
The word ‘isosceles’ is derived from the Latin word “īsoscelēs’ and the ancient Greek word ‘ἰσοσκελής (isoskelḗs)’ which means ‘equal-legged.’
The Babylonian and Egyptian mathematics knew how to calculate ‘area’ much before the ancient Greek mathematicians studied the isosceles triangle.
If a building is shaped like isosceles, it not only makes them attractive but also earthquake resistant.
The shapes of this triangle are often used in construction due to their high strength.
Always remember that the sum of the three angles of the isosceles triangle is always 180 degrees. So, if you know the value of two angles, finding the value of the third angle is easy.
FAQs on Isosceles Triangle Theorems and Key Properties
1. What is an isosceles triangle?
An isosceles triangle is a triangle with two equal sides and two equal angles opposite those sides. The equal sides are called legs, and the third side is called the base.
- The angles opposite the equal sides are called base angles.
- The vertex angle is the angle between the two equal sides.
- It is a special case of a triangle studied under isosceles triangle theorems.
2. What is the Isosceles Triangle Theorem?
The Isosceles Triangle Theorem states that if two sides of a triangle are equal, then the angles opposite those sides are equal. In other words:
- If AB = AC, then ∠B = ∠C.
- This explains why base angles in an isosceles triangle are congruent.
- It is one of the most important geometry theorems related to triangle properties.
3. What is the converse of the Isosceles Triangle Theorem?
The converse of the Isosceles Triangle Theorem states that if two angles of a triangle are equal, then the sides opposite those angles are equal. This means:
- If ∠B = ∠C, then AB = AC.
- This helps prove that a triangle is isosceles when given equal angles.
- It is often used in geometric proofs.
4. How do you find the base angles of an isosceles triangle?
To find the base angles of an isosceles triangle, subtract the vertex angle from 180° and divide by 2. Since the sum of angles in a triangle is 180°:
- Base angle = (180° − vertex angle) ÷ 2
- Example: If vertex angle = 40°, then base angles = (180 − 40) ÷ 2 = 70° each.
5. What is the formula for the area of an isosceles triangle?
The area of an isosceles triangle is calculated using Area = (1/2) × base × height. To find the height when the equal sides are known:
- Height = √(a² − (b/2)²), where a = equal side and b = base.
- Then substitute into the area formula.
- Example: If a = 5 and b = 6, height = √(25 − 9) = √16 = 4, so area = (1/2) × 6 × 4 = 12 square units.
6. Why are the base angles equal in an isosceles triangle?
The base angles are equal in an isosceles triangle because of the Isosceles Triangle Theorem, which states that equal sides have equal opposite angles. Since two sides are congruent:
- Each side faces an angle of the same measure.
- This symmetry ensures the triangle has two equal base angles.
- This property is fundamental in triangle geometry proofs.
7. How do you prove a triangle is isosceles?
You prove a triangle is isosceles by showing that either two sides are equal or two angles are equal. Common proof methods include:
- Using the converse of the Isosceles Triangle Theorem.
- Applying congruence rules like SSS, SAS, or ASA.
- Showing perpendicular bisectors create equal segments.
8. What is the perimeter of an isosceles triangle?
The perimeter of an isosceles triangle is the sum of all three sides, calculated as P = 2a + b, where a is the equal side and b is the base. For example:
- If equal sides = 5 cm and base = 8 cm,
- P = 2(5) + 8 = 18 cm.
9. What is the difference between an isosceles and an equilateral triangle?
The main difference is that an isosceles triangle has two equal sides, while an equilateral triangle has three equal sides. Key differences include:
- Isosceles: two equal sides and two equal angles.
- Equilateral: three equal sides and three equal angles of 60° each.
- An equilateral triangle is a special case of an isosceles triangle.
10. What happens to the median in an isosceles triangle?
In an isosceles triangle, the median drawn from the vertex to the base is also the altitude and the perpendicular bisector of the base. This means:
- It divides the base into two equal parts.
- It forms a 90° angle with the base.
- It creates two congruent right triangles.
