# Class 9 Math Sides and Angles of Triangle Example \mathbf{A B C} is an isosceles triangle with base \overline{\mathbf{B C}} \cdot On \overrightarrow{\mathbf{B C}} a point \mathrm{D} is taken away from C. A line segment through D cuts

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##### Example \mathbf{A B C} is an isosceles triangle with base \overline{\mathbf{B C}} \cdot On \overrightarrow{\mathbf{B C}} a point \mathrm{D} is taken away from C. A line segment through D cuts \overline{\mathbf{A C}} at \mathbf{L} and \overline{\mathbf{A B}} at M . Prove that \mathrm{m} \overline{\mathbf{A L}}>\mathrm{m} \overline{\mathbf{A M}} .

6. If 3 \mathrm{~cm} and 4 \mathrm{~cm} are lengths of two sides of a right angle triangle then what should be the third length of the triangle.

4. Prove that in a right-angled triangle the hypotenuse is longer than each of the other two sides.

1. Two sides of a triangle measure 10 \mathrm{~cm} and 15 \mathrm{~cm} . Which of the following measure is possible for the third side?(b) 20 \mathrm{~cm}

1. In the figure P is any point and A B is a line. Which of the following is the shortest distance between the point \mathrm{P} and the line \mathrm{AB} ?(c) \mathrm{mPN}

3. In the \triangle \mathrm{ABC} \mathrm{m} \angle \mathrm{B}=70^{\circ} and \mathrm{m} \angle \mathrm{C}=45^{\circ} . Which of the sides of the triangle is longest and which is the shortest?

4. \cdots If 10 \mathrm{~cm} 6 \mathrm{~cm} and 8 \mathrm{~cm} are the lengths of a triangle then verify that sum of measures of two sides of a triangle is greater than the third side.

Example 1Which of the following sets of lengths can be the lengths of the sides of a triangle?(a) 2 \mathrm{~cm} 3 \mathrm{~cm} 5 \mathrm{~cm}

1. Two sides of a triangle measure 10 \mathrm{~cm} and 15 \mathrm{~cm} . Which of the following measure is possible for the third side?(c) 25 \mathrm{~cm}

1. Two sides of a triangle measure 10 \mathrm{~cm} and 15 \mathrm{~cm} . Which of the following measure is possible for the third side?(d) 30 \mathrm{~cm}

3. In the figure \overline{\mathrm{PL}} is perpendicular

5. 3 \mathrm{~cm} 4 \mathrm{~cm} and 7 \mathrm{~cm} are not the lengths of the triangle. Give the reason.

3. If 13 \mathrm{~cm} 12 \mathrm{~cm} and 5 \mathrm{~cm} are the lengths of a triangle then verify that difference of measures of any two sides of a triangle is less than the measure of the third side.

Example 1Prove that in a scalene triangle the angle opposite to the largest side is of measure greater than \mathbf{6 0}^{\circ} . (i.e. two-third of a right-angle)

Which of the following are true ard which are false?(i) The angle opposite to the longer side of a triangle is greater.(ii) In a right-angled triangle greater angle is of 60^{\circ} .(iii) In an isosceles right-angled triangle angles other than right angle are each of 45^{\circ} .(iv) A triangle having two congruent sides is called equilateral triangle.(v) A perpendicular from a point to a line is shortest distance.(vi) Perpendicular to line form an angle of 90^{\circ} .(vii) A point out side the line is collinear with it.(viii) Sum of two sides of triangle is greater than the third.(ix) The distance between a line and a point on it is zero.(x) Triangle can be formed of lengths 2 \mathrm{~cm} 3 \mathrm{~cm} and 5 \mathrm{~cm} .

2. In the figure \mathrm{P} is any point lying away from the line \mathrm{AB} . Then \mathrm{m} \overline{\mathrm{PL}} will be the shortest distance if(b) \mathrm{m} \angle \mathrm{PEB}=100^{\circ}

4. Prove that in a right-angled triangle the hypotenuse is longer than each of the other two sides.5. In the triangular figure \mathrm{m} \overline{\mathrm{AB}}>\mathrm{m} \overline{\mathrm{AC}} \cdot \overline{\mathrm{BD}} and \overline{\mathrm{CD}} are the bisectors of \angle \mathrm{B} and \angle \mathrm{C}

1. In the figure P is any point and A B is a line. Which of the following is the shortest distance between the point \mathrm{P} and the line \mathrm{AB} ?(b) \mathrm{mPM}

Example 1Which of the following sets of lengths can be the lengths of the sides of a triangle?(c) 2 \mathrm{~cm} 4 \mathrm{~cm} 7 \mathrm{~cm}

1. In the figure P is any point and A B is a line. Which of the following is the shortest distance between the point \mathrm{P} and the line \mathrm{AB} ?(a) \mathrm{mPL}

2. In the figure P is any point lying away from the line \mathrm{AB} . Then \mathrm{mPL} will be the shortest distance if(c) \mathrm{m} \angle \mathrm{PLA}=90^{\circ}

2. In the figure P is any point lying away from the line \mathrm{AB} . Then \mathrm{m} \overline{\mathrm{PL}} will be the shortest distance if(a) m \angle P L A=80^{\circ}

Example 1Which of the following sets of lengths can be the lengths of the sides of a triangle?(b) 3 \mathrm{~cm} 4 \mathrm{~cm} 5 \mathrm{~cm}

2. What will be angle for shortest distance from an outside point to the line?

1. In the figure P is any point and A B is a line. Which of the following is the shortest distance between the point \mathrm{P} and the line \mathrm{AB} ?(d) \mathrm{m} \overline{\mathrm{PO}}

1. Two sides of a triangle measure 10 \mathrm{~cm} and 15 \mathrm{~cm} . Which of the following measure is possible for the third side?(a) 5 \mathrm{~cm}

Example 2In a rilateral \mathbf{A B C D} \overline{\mathbf{A B}} is the longest side and \overline{\mathbf{C D}} is the shortest side. Prove that \mathbf{m} \angle \mathbf{B C D}>\mathbf{m} \angle \mathbf{B A D} .