Properties of Isosceles Triangle
A triangle with two equal sides is called an isosceles triangle. The word 'isosceles' comes from Greek: 'iso' means equal and 'skelos' means leg.
The two equal sides are called the legs, and the third side is called the base. The angle between the two equal sides is the vertex angle, and the two angles opposite the equal sides are called base angles.
The most important property: the base angles of an isosceles triangle are equal. This is one of the earliest theorems you learn in geometry.
What is Properties of Isosceles Triangle - Grade 7 Maths (Triangles)?
Definition: An isosceles triangle is a triangle that has at least two sides of equal length.
Parts of an isosceles triangle:
- Legs: The two equal sides.
- Base: The unequal side (opposite the vertex angle).
- Vertex angle: The angle between the two equal sides.
- Base angles: The two angles adjacent to the base.
Properties of Isosceles Triangle Formula
Key Property (Isosceles Triangle Theorem):
If two sides of a triangle are equal, the angles opposite to them are also equal.
Converse:
If two angles of a triangle are equal, the sides opposite to them are also equal.
Using Angle Sum Property:
- If vertex angle = A and base angles = B each, then: A + B + B = 180°
- A + 2B = 180°
- B = (180° − A) / 2
Types and Properties
Special Cases:
- Right isosceles triangle: vertex angle = 90°, base angles = 45° each.
- Equilateral triangle: A special isosceles triangle where ALL three sides are equal (and all angles are 60°).
- Obtuse isosceles triangle: vertex angle > 90°, base angles are acute.
- Acute isosceles triangle: All three angles are acute, with two of them equal.
Solved Examples
Example 1: Finding Base Angles
Problem: An isosceles triangle has a vertex angle of 40°. Find the base angles.
Solution:
- Let each base angle = x
- 40 + x + x = 180
- 40 + 2x = 180
- 2x = 140
- x = 70
Answer: Each base angle is 70°.
Example 2: Finding Vertex Angle
Problem: The base angles of an isosceles triangle are 55° each. Find the vertex angle.
Solution:
- Vertex angle = 180° − 55° − 55° = 70°
Answer: The vertex angle is 70°.
Example 3: Using the Converse
Problem: In triangle PQR, ∠Q = ∠R = 65°. Is PQ = PR?
Solution:
- Two angles are equal (∠Q = ∠R).
- By the converse of the isosceles triangle theorem, the sides opposite equal angles are equal.
- Side opposite ∠Q is PR. Side opposite ∠R is PQ.
Answer: Yes, PQ = PR. Triangle PQR is isosceles.
Example 4: Right Isosceles Triangle
Problem: An isosceles triangle has a right angle. Find all angles.
Solution:
- The right angle (90°) must be the vertex angle.
- Base angles: (180 − 90)/2 = 45° each.
Answer: Angles are 90°, 45°, 45°.
Example 5: Perimeter Problem
Problem: An isosceles triangle has equal sides of 8 cm each and base of 6 cm. Find its perimeter.
Solution:
- Perimeter = 8 + 8 + 6 = 22 cm
Answer: Perimeter = 22 cm.
Example 6: Finding Side Length
Problem: The perimeter of an isosceles triangle is 34 cm. The base is 10 cm. Find the length of each equal side.
Solution:
- Let each equal side = x
- x + x + 10 = 34
- 2x = 24
- x = 12
Answer: Each equal side = 12 cm.
Example 7: Using Variables
Problem: In an isosceles triangle, the vertex angle is (2x + 10)° and each base angle is (x + 25)°. Find all angles.
Solution:
- (2x + 10) + (x + 25) + (x + 25) = 180
- 4x + 60 = 180
- 4x = 120
- x = 30
- Vertex = 70°, Base angles = 55° each
- Check: 70 + 55 + 55 = 180 ✓
Answer: Angles: 70°, 55°, 55°.
Example 8: Altitude as Line of Symmetry
Problem: In isosceles triangle ABC (AB = AC), the altitude from A meets BC at D. Show that BD = DC.
Solution:
- The altitude from the vertex angle to the base in an isosceles triangle bisects the base.
- This is because the altitude is also the perpendicular bisector and the median.
- So BD = DC.
Answer: BD = DC. The altitude from the vertex bisects the base.
Real-World Applications
Real-world examples:
- Roof trusses: Many roofs are shaped as isosceles triangles for symmetric load distribution.
- Traffic signs: Yield signs and some warning signs are isosceles triangles.
- Architecture: Pediments on Greek temples are isosceles triangles.
- Paper folding: Folding a rectangular paper diagonally creates an isosceles right triangle.
Key Points to Remember
- An isosceles triangle has two equal sides (legs) and one unequal side (base).
- Base angles are equal — angles opposite the equal sides.
- Converse: if two angles are equal, the opposite sides are equal.
- The altitude from the vertex angle bisects the base and is also the median and angle bisector.
- An isosceles triangle has 1 line of symmetry.
- An equilateral triangle is a special case of an isosceles triangle.
- Vertex angle + 2 × base angle = 180°.
Practice Problems
- An isosceles triangle has base angles of 72° each. Find the vertex angle.
- The vertex angle of an isosceles triangle is 100°. Find each base angle.
- Perimeter of an isosceles triangle is 40 cm. Base = 12 cm. Find equal sides.
- Can an isosceles triangle have angles 60°, 60°, 60°? What is it called?
- In an isosceles triangle, one angle is 120°. Find the other two angles.
- The equal sides of an isosceles triangle are 10 cm and the base is 8 cm. An altitude from the vertex to the base is drawn. Find the length of each half of the base.
Frequently Asked Questions
Q1. What is an isosceles triangle?
A triangle with at least two sides of equal length. The equal sides are called legs and the third side is the base.
Q2. Are the base angles always equal?
Yes. In an isosceles triangle, the angles opposite the equal sides are always equal. This is the Isosceles Triangle Theorem.
Q3. Can an isosceles triangle be right-angled?
Yes. If the vertex angle is 90°, the two base angles are 45° each. This is called a right isosceles triangle.
Q4. Is an equilateral triangle also isosceles?
Yes. An equilateral triangle has all three sides equal, which means any two sides are equal. So it satisfies the definition of an isosceles triangle.
Q5. How many lines of symmetry does an isosceles triangle have?
Exactly 1 — the line from the vertex angle to the midpoint of the base.
Related Topics
- Properties of Equilateral Triangle
- Angle Sum Property of Triangle
- Classification of Triangles
- Congruence of Triangles
- Exterior Angle Property of Triangle
- Triangle Inequality Property
- Medians and Altitudes of Triangle
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
- Criteria for Similarity of Triangles
