Triangle Theorems
Triangles are governed by two important inequalities. The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. This note contains information about triangle theorem.
Summary
Triangles are governed by two important inequalities. The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. This note contains information about triangle theorem.
Things to Remember
- Triangles are governed by two important inequalities.
- A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two.
- The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
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Triangle Theorems
Triangles are governed by two important inequalities. The first is often referred to as the triangle inequality. It states that the length of a side of a triangle is always less than the sum of the lengths of the other two sides.
The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
Theorem 1
The sum of interior angles of a triangle is 180°
Draw three different triangles in your notebook. Measure ∠X, ∠Y and ∠Z using a protector and fill in the table.
Verification:
| Figure | ∠X | ∠Y | ∠Z | ∠X +∠Y+∠Z |
| (i) | ||||
| (ii) | ||||
| (iii) |
Look at the figure and complete the table given below.
| Statements | Reasons |
| a+b+c = 180° | Sum of adjacent angles on a straight line |
| a = m, c = n | Corresponding angles |
| m+b+n = 180° | ? |
Conclusion: The sum of interior angles of a triangle is 180°
Theorem 2
Base angles of an isosceles triangle are equal.
Draw three different triangles making AB = AC, ∠B and ∠C opposite to AC and AB respectively are the base angles. Measure ∠ABC and ∠ACB using a protector and fill in the table.
Verification:
| Figure | ∠ABC | ∠ACB | Result |
| (i) | ∠ABC =∠ACB | ||
| (ii) | |||
| (iii) |
Conclusion: Base angles of an isosceles triangle are equal.
Theorem 3
Each of the base angles of an isosceles right triangle is 45°.
Draw three triangles making ∠B = 90° and AC = BC. Measure ∠BAC and ∠ACB and fill in the table.
Verification:
| Figure | ∠BAC | ∠ACB | Result |
| (i) | ∠BAC =∠ACB = 45° | ||
| (ii) | |||
| (iii) |
Conclusion: Each of the base angles of an isosceles right triangle is 45°
Theorem 4
The line joining the vertex and midpoint of the base of an isosceles triangle is perpendicular to the base.
Draw three triangles making AB = AC. Join the midpoint P of BC and A, in each figure. Measure the angles APB and APC and fill in the blanks.
Verification:
| Figure | ∠APB | ∠APC | Result |
| (i) | ∠APB =∠APC =90° | ||
| (ii) | |||
| (iii) |
Conclusion: The line joining the vertex and mid-point of the base of an isosceles triangle is perpendicular to the base.
Theorem 5
All the angles of an equilateral triangle are equal.
Draw three triangles making AB = BC =CA in each figure. Measure∠ABC, ∠BCA and ∠CAB and fill in the table given below.
Verification:
| Figure | ∠ABC | ∠BCA | ∠CAB | Result |
| (i) | ∠ABC =∠BCA =∠CAB | |||
| (ii) | ||||
| (iii) |
Conclusion: All angles of an equilateral triangle are equal.
Lesson
Geometry
Subject
Compulsory Maths
Grade
Grade 8
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