Definition: The figure bounded by three line segments is a triangle
1.The figure bounded by three line segments is a triangle
2.The line segments are known as the sides of the triangle.
3.The point common to any two sides is known as vertex. The angle formed at the vertex is an angle of the triangle.
4.The triangle has three sides and three angles.
*** Depending on the lengths of the sides the triangles are of three types:
i) equilateral, ii) Isosceles and iii) Scalene.
***On the basis of the angles, the triangles are also of three types:
i) acute angled, ii) Obtuse angled and iii) Right angled.
5.The sum of the lengths of the sides is called perimeter of the triangle.
6. Medians of a triangle
A median connects a vertex of a triangle to the midpoint of the opposite side.
Here AD is median of triangle ABC
7. Altitudes of a triangle
An altitude has one end point at a vertex of the triangle and the other on the
line containing the opposite side
- Exterior and interior angles of a triangle
If any side of a triangle is extended the angle formed is an exterior angle. The two angles other than the adjacent interior angle are known as the interior opposite angles. In the ad joint figure, the side BC of ΔABC is produced to D. Observe the angle ∠ACD formed at the point C. This angle lies in the exterior of ∠ABC. We call it an exterior angle of the ∠ABC formed at vertex C. Clearly ∠ACB is an adjacent angle to ∠ACD. The remaining two angles of the triangle namely ∠ABC and
∠BAC are called the two interior opposite angles or the two remote interior angles of ∠ACD.
9.The sum of the three angles of a triangle is equal to two right angles.
Some corollary about triangle:
Corollary i: If a side of a triangle is produced then exterior angle so formed is equal to the sum of the two opposite interior angles. .
Corollary ii: If a side of a triangle is produced then the exterior angle so formed is greater than each of the two interior opposite angles.
Corollary iii: The acute angles of a right angled triangle are complementary to each other.
Corollary iv: In an equilateral triangle each angle measures 60°.
11.If two sides of a triangle are equal, their opposite angles are also equal..
****The sum of the angles of a quadrilateral is equal to 4 right angles.
***If one side of a triangle is greater than another, then the angle opposite the greater side is greater than the angle opposite the smaller side
***If one angle of a triangle is greater than another, then the side opposite the greater angle is greater than the side opposite the smaller angle.
*** The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Answer the questions 1-3 on the basis of the following information:
In the figure, the side BC of the triangle ABC is extended to D. CE is the bisector of
∠ACD. AB || CE and ∠ECD = 60°.
- Which one of the following is the value of ∠BAC?
(a) 30° (b) 45° (c) 60° (d) 120°
- Which one of the following is the value of ∠ACD?
(a) 60° (b) 90° (c) 120° (d) 180°
- What type of triangle ΔABC is?
(a) obtuse-angled (b) isosceles (c) equilateral (d) right-angled
- If ∠A=70° and ∠B = 40° in ΔABC then what type of triangle isΔABC ?
(a) obtuse-angled (b) right–angled (c) equilateral (d) isosceles.
5 The lengths of two sides of a triangle are 5 cm. and 4 cm Which one of the following is the possible measurement of the third side of the triangle?
(a) 1 cm (b) 4 cm (c) 9 cm (d) 10 cm
- The equal sides of an isosceles triangle are extended and if one of the external angles is 120° then how much is the other external angle?
(a) 120° (b) 90° (c) 60° (d) 30°
- If one of the two acute angles of a right angled triangle is 40°, then which of the following is the value of other acute angle?
(a) 40° (b) 45° (c) 50° (d) 60°
- If the sum of two angles is equal to the third angle of a triangle, what type of triangle is it?
(a) equilateral (b) acute-angled (c) right-angled (d) obtuse-angled
- If two sides of a triangle and one angle opposite to one of the sides are given,
then how many triangles can be drawn?
(a) 1 (b) 2 (c) 3 (d) 4
- In which case is it possible to draw a triangle when the lengths of the sides are
(a) 1 cm, 2 cm, 3 cm (b) 3 cm. 4 cm, 5 cm.
(c) 2 cm, 4 cm, 6 cm. (d) 3 cm, 4 cm, 7 cm.
- (i) If two sides of a triangle and the included angle are given then the triangle can be drawn.
(ii) If the sum of two sides of a triangle is greater than the third side then the triangle can be drawn.
(iii) There may exist more then one obtuse angle in a triangle.
Which one of the following is correct?
(a) i and ii (b) ii and iii (c) i and iii (d) i, ii and iii.