Chapter 7: A Tale of Three Intersecting Lines
Three lines meeting in pairs make a triangle, the simplest closed shape. In this chapter we construct triangles with a compass and protractor, find out which side lengths and angles are possible, prove that the angles always add up to 180°, draw altitudes, and sort triangles by their sides and angles.
In this chapter you will
- construct triangles from sides, or from sides and angles
- use the triangle inequality to test side lengths
- use the angle sum property and exterior angles
- draw altitudes and classify triangles
Topics
- Equilateral Triangles NCERT 7.1
Equilateral: all three sides equal; isosceles: two sides equal.
- Constructing a Triangle When its Sides are Given NCERT 7.2
Given three sides: base, then two arcs from its ends; they cross at the third vertex.
- When Some Sides and Angles are Given NCERT 7.3
A triangle can be built from two sides and the included angle, or two angles and the included side.
- Altitudes of Triangles NCERT 7.4
An altitude is the perpendicular from a vertex to the opposite side (extended if needed).
- Types of Triangles NCERT 7.5
By sides: equilateral, isosceles, scalene. By angles: acute-, right-, obtuse-angled.
Key words
- Equilateral
- all three sides equal
- Isosceles
- two sides equal
- Scalene
- all sides different
- Triangle inequality
- each side is less than the sum of the other two
- Angle sum property
- the angles of a triangle add up to 180°
- Exterior angle
- angle between one side and the extension of another
- Altitude
- perpendicular from a vertex to the opposite side
Summary
- With a compass, the third vertex is where two arcs from the base ends cross.
- Three lengths make a triangle exactly when the longest is less than the sum of the other two.
- The angles of a triangle add up to 180°; two angles must add to less than 180°.
- Altitudes are perpendiculars from the vertices; triangles are named by sides and by angles.