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Q. Find the unknown length x in the following figures (Fig \small 6.29):

    
 

As we know in a Right-angled Triangle: By Pythagoras Theorem, So, using this theorem, i)  . ii) iii) iv) v) in this question as we can see from the figure, it is making the right angle with the half-length of x, so . vi)

2.  Find angles x and y in each figure.

i) As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal. So, Now,  As we know the sum of internal angles of a triangle is 180. so, Hence, . ii)  As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal. AND the sum of internal angles of a triangle is 180. so, Also, . Hence . iii) As we...

1. Find angle x in each figure: 

        

As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal. and the sum of angles of the triangle is equal to . So, i)  ii)  iii)  iv)  v) vi) vii) viii) As we know, the exterior angle is equal to the sum of opposite internal angles in a triangle. So,  ix)As we know when two lines are intersecting, the opposite angles are equal. So

3.  Is something wrong in this diagram (Fig \small 6.12)? Comment.
 

            

Yes, The measure of the exterior angle is given wrong. As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles. So, Exterior angle =                         =  Hence the exterior angle should be equal be  instead of .

1.  Exterior angles can be formed for a triangle in many ways. Three of them are shown here (Fig  \small 6.10)

There are three more ways of getting exterior angles. Try to produce those rough sketches.

In a triangle, there are a total of six exterior angles Exterior Angles in a Triangle  

2. Draw rough sketches of altitudes from A to \small \overline{BC} for the following triangles (Fig \small 6.6):

        

altitudes from A to for the triangles are:

3.  Look at Fig \small 6.2 and classify each of the triangles according to its

            (a) Sides

            (b) Angles

         

i) The triangle ABC Based on Side: In Triangle ABC, Since two sides (BC and AC ) are equal (= 8 cm ) The given triangle is an isosceles triangle. Based on Angle: In Triangle ABC, Since all the triangles are less than 90 degrees, So the given triangle is Acute angled triangle.   ii) The Triangle PQR Based on Side: In Triangle PQR, All the sides are different so, The given triangle is a...

2.  Write the:

            (i) Side opposite to the vertex Q of \small \Delta PQR

            (ii) Angle opposite to the side LM of \small \Delta LMN

            (iii) Vertex opposite to the side RT of \small \Delta RST

(i) The side opposite to the vertex Q of               (ii) Angle opposite to the side LM of      (iii) Vertex opposite to the side RT of   = Vertex S.

Q : 1    Write the six elements (i.e., the 3 sides and the 3 angles) of  \small \Delta ABC

The Triangle  .    The Elements of the triangle are: Sides :  Angles :   

2. Does a median lie wholly in the interior of the triangle? (If you think that this is not true, draw a figure to show such a case).
 

Yes, The median always lies in the interior of the triangle. As we can see in all three cases the median lies inside the triangle.

1. How many medians can a triangle have?

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex.

5.  Can the altitude and median be same for a triangle?

            (Hint: For Q.No.  4 and 5, investigate by drawing the altitudes for every type of triangle).

Yes, the altitude and median can be the same in a triangle. for example, consider an equilateral triangle, the median which divides the side in equal is also perpendicular to the side and hence the altitude and the median is the same.  

4. Can you think of a triangle in which two altitudes of the triangle are two of its sides? 

Yes, in a Right - angled triangle, the two altitudes of the sides are two sides as they make an angle of 90 degrees with one another.  

3. Will an altitude always lie in the interior of a triangle? If you think that this need not be true, draw a rough sketch to show such a case. 

No, the altitude of a triangle might lie outside the triangle. for example in the obtuse-angled triangle, we have to extend the base side for making altitude angle.  

1. How many altitudes can a triangle have?
 

Every triangle has three bases (any of its sides) and three altitudes (heights). Every altitude is the perpendicular segment from a vertex to its opposite side (or the extension of the opposite side.

3. What can you say about the sum of an exterior angle of a triangle and its adjacent interior angle?
 

As we can see they both angles forms a straight line, Hence the sum of an exterior angle of a triangle and its adjacent interior angle is always .

2. Are the exterior angles formed at each vertex of a triangle equal?

Yes, why because in each vertex there are two exterior angles are there and those are the vertically opposite angles also. therefore those are equal.

2. The two interior opposite angles of an exterior angle of a triangle are \small 60^{\circ} and \small 80^{\circ}. Find the measure of the exterior angle. 

As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles. So, Exterior Angle =  + .                         =  Hence the measure of the exterior is .

1. An exterior angle of a triangle is of measure \small 70^{\circ} and one of its interior opposite angles is of measure \small 25^{\circ}. Find the measure of the other interior opposite angle. 

As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles. So, According to the question, Hence the other interior angle is .

2. Can the exterior angle of a triangle be a straight angle?
 

No, the exterior angle of a triangle cannot be a straight angle because if the exterior angle is straight then there won't be any triangle left that would be a straight line. (imagine it visually).
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