Key Properties and Real-Life Applications of Isosceles Triangles
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: Definitions, Properties & Key Proofs
1. What are the two main theorems for an isosceles triangle as per the CBSE syllabus?
The two fundamental theorems for isosceles triangles, crucial for Class 9 Maths, are:
- Isosceles Triangle Theorem: It states that if two sides of a triangle are equal, then the angles opposite to those sides are also equal.
- Converse of the Isosceles Triangle Theorem: It states the reverse: if two angles of a triangle are equal, then the sides opposite to those angles are also equal.
2. What are the key properties of an isosceles triangle?
An isosceles triangle has several distinct properties:
- It has two equal sides (called legs) and one unequal side (called the base).
- The two angles opposite the equal sides (base angles) are equal.
- The sum of all its interior angles is always 180 degrees.
- The altitude drawn from the vertex angle (the angle between the two equal sides) to the base bisects the base and the vertex angle.
- This same altitude is also the perpendicular bisector of the base and the median to the base.
3. How do you prove the Isosceles Triangle Theorem (angles opposite to equal sides are equal)?
To prove that angles opposite to equal sides in a triangle are equal, you can use the Side-Angle-Side (SAS) congruence rule. Consider a triangle ABC where AB = AC. To prove ∠B = ∠C, you would follow these steps:
- Construction: Draw an angle bisector of ∠A, and let it intersect the base BC at point D.
- Proof: Now, compare ΔABD and ΔACD.
- 1. AB = AC (Given)
- 2. ∠BAD = ∠CAD (By construction, as AD is the angle bisector)
- 3. AD = AD (Common side)
- Therefore, ΔABD ≅ ΔACD by the SAS congruence rule.
- Since the triangles are congruent, their corresponding parts are equal. Hence, ∠B = ∠C.
4. How can you determine if a given triangle is isosceles just by its angles?
You can determine if a triangle is isosceles by applying the converse of the Isosceles Triangle Theorem. The theorem states that if two angles in a triangle are equal, then the sides opposite those angles are also equal. For instance, in a triangle PQR, if you measure or are given that ∠Q = ∠R, you can conclude that the sides opposite to them, PR and PQ, must be equal. This makes ΔPQR an isosceles triangle.
5. What is the primary difference between an isosceles triangle and an equilateral triangle?
The main difference lies in the number of equal sides and angles.
- An isosceles triangle has a minimum of two equal sides and two equal angles.
- An equilateral triangle has all three sides equal and all three angles equal (each being 60°).
Therefore, every equilateral triangle is a special type of isosceles triangle, but not all isosceles triangles are equilateral.
6. Can an isosceles triangle also be a right-angled triangle?
Yes, an isosceles triangle can be a right-angled triangle. This special type of triangle is called an isosceles right-angled triangle. In such a triangle, one angle is 90°, and the other two angles must be equal. Since the sum of angles in a triangle is 180°, the two equal angles would each be (180° - 90°) / 2 = 45°. The sides opposite these 45° angles are equal.
7. What is a common mistake students make when applying isosceles triangle theorems?
A common mistake is incorrectly identifying the corresponding sides and angles. Students might assume that any two angles are equal without checking if they are opposite the two equal sides. For example, in a triangle ABC where AB = AC, the equal angles are ∠B and ∠C. It is incorrect to assume ∠A is equal to ∠B or ∠C unless the triangle is also equilateral. Always match the equal angles with their opposite equal sides.
8. Where can we see examples of isosceles triangles in real-world structures?
Isosceles triangles are fundamental to design and engineering for their stability. You can find them in:
- Architecture: The gables of many houses and the triangular trusses used in bridges and roofs are often isosceles.
- Everyday Objects: A common clothes hanger, a slice of pizza, or the tip of a pair of scissors often form an isosceles triangle.
- Geometry Tools: One of the triangles in a standard geometry set (the 45°-45°-90° set square) is an isosceles right-angled triangle.
