Theorem 7.8 : The sum of any two sides of a triangle is greater than the third side.
Text Solution
AI Generated Solution
To prove Theorem 7.8: The sum of any two sides of a triangle is greater than the third side, we will follow these steps:
### Step-by-Step Solution:
1. **Draw Triangle ABC**:
Begin by sketching a triangle labeled ABC.
2. **Extend Side AB**:
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Which of the following statements are true (T) and which are false (F)? (i) Sum of the three sides of a triangle is less than the sum of its three altitudes. (ii) Sum of any two sides of a triangle is greater than twice the median drawn to the third side. (iii) Sum of any two sides of a triangle is greater than the third side. (iv) Difference of any two sides of a triangle is equal to the third side. (v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it. (vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.
The sum of any two sides of a triangle is greater than the third side. GIVE : A triangle A B C TO PROVE : A B+A C > B C ,A B+B C > A C and B C+A C > A B CONSTRUCTION : Produce side B A to D such that A D=A Cdot Join C D .
Fill in the blanks to make the following statements true: In a right triangle the hypotenuse is the ...... side. The sum of three altitudes of a triangle is ......... than its perimeter. The sum of any two sides of a triangle is ......... than the third side. If two angles of a triangle are unequal, then the smaller angle has the ........... side opposite to it. Difference of any two sides of a triangle is ......... than the third side. If two sides of a triangle are unequal, then the larger side has ..... angle opposite to it.
Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.
Show that the difference of any two sides of a triangle is less than the third side.
Show that the difference of any two sides of a triangle is less than the third side. GIVEN : triangle A B C TO PROVE : (i) AC-AB < BC
Using Theorem 6.2, prove that the line joining the mid-point of any two sides of a triangle is parallel to the third side. (Recall that you have done it in class IX).
The line segment joining the mid-points of any two sides of a triangle in parallel to the third side and equal to half of it.
Prove that the line joining the mid-points of the two sides of a triangle is parallel to the third side.
Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
