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4. Find the value of unknowns for the given congruent triangles.

### 4. Find the value of unknowns for the given congruent triangles.

2. From a point on the line bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

### 2. From a point on the line bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

1. \mathrm{ABC} is an isosceles triangle. \mathrm{D} is the mid-point of base \overline{\mathrm{BC}} . Prove that \overline{\mathrm{AD}} bisects \angle \mathrm{A} and \overrightarrow{\mathrm{AD}} \perp \overrightarrow{\mathrm{BC}} .

### 1. \mathrm{ABC} is an isosceles triangle. \mathrm{D} is the mid-point of base \overline{\mathrm{BC}} . Prove that \overline{\mathrm{AD}} bisects \angle \mathrm{A} and \overrightarrow{\mathrm{AD}} \perp \overrightarrow{\mathrm{BC}} .

2. Prove that a point which is equidistant from the end points of a line segment is on the right bisector of the line segment.

### 2. Prove that a point which is equidistant from the end points of a line segment is on the right bisector of the line segment.

3. Fill in the blanks to make the sentences true sentences:(vi) The sum of the measures of acute angle of a right triangle is

### 3. Fill in the blanks to make the sentences true sentences:(vi) The sum of the measures of acute angle of a right triangle is

(iv) How many acute angles are there in an acute angled triangle?(a) 1(b) 2(c) 3(d) not more than 2 .

### (iv) How many acute angles are there in an acute angled triangle?(a) 1(b) 2(c) 3(d) not more than 2 .

3. Fill in the blanks to make the sentences true sentences:(v) In a right-angled triangle side opposite to right angle is called

### 3. Fill in the blanks to make the sentences true sentences:(v) In a right-angled triangle side opposite to right angle is called

1. In the figure \overline{\mathrm{AB}} \cong \overline{\mathrm{DC}} \overline{\mathrm{AD}} \cong \overline{\mathrm{BC}} . Prove that \angle \mathrm{A} \cong \angle \mathrm{C} \angle \mathrm{ABC} \cong \angle \mathrm{ADC} .

### 1. In the figure \overline{\mathrm{AB}} \cong \overline{\mathrm{DC}} \overline{\mathrm{AD}} \cong \overline{\mathrm{BC}} . Prove that \angle \mathrm{A} \cong \angle \mathrm{C} \angle \mathrm{ABC} \cong \angle \mathrm{ADC} .

1. In the given figure m \overline{\mathrm{AB}}=m \overline{\mathrm{CB}} and \angle \mathrm{A} \cong \angle \mathrm{C} prove that \triangle \mathrm{ABD} \cong \triangle \mathrm{CBE}

### 1. In the given figure m \overline{\mathrm{AB}}=m \overline{\mathrm{CB}} and \angle \mathrm{A} \cong \angle \mathrm{C} prove that \triangle \mathrm{ABD} \cong \triangle \mathrm{CBE}

3. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=25^{\circ} m \angle \mathrm{B}=45^{\circ} and \overline{\mathrm{CD}} \perp \overline{\mathrm{AB}} . Prove that \triangle \mathrm{DBC} is an isosceles \Delta .

### 3. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=25^{\circ} m \angle \mathrm{B}=45^{\circ} and \overline{\mathrm{CD}} \perp \overline{\mathrm{AB}} . Prove that \triangle \mathrm{DBC} is an isosceles \Delta .

3. P Q R S is a square. X Y and Z are the mid-points of \overline{P Q} \overline{Q R} and \overline{\mathrm{RS}} respectively. Prove that \triangle \mathrm{PXY} \cong \Delta \mathrm{SZY} .

### 3. P Q R S is a square. X Y and Z are the mid-points of \overline{P Q} \overline{Q R} and \overline{\mathrm{RS}} respectively. Prove that \triangle \mathrm{PXY} \cong \Delta \mathrm{SZY} .

(ii) In \triangle \mathrm{ABC} if \angle \mathrm{A} \cong \angle \mathrm{B} then the bisector of angle divides the triangle into congruent triangles:(a) \angle \mathrm{A} (b) \angle B (c) \angle C (d) any one of its angles.

### (ii) In \triangle \mathrm{ABC} if \angle \mathrm{A} \cong \angle \mathrm{B} then the bisector of angle divides the triangle into congruent triangles:(a) \angle \mathrm{A} (b) \angle B (c) \angle C (d) any one of its angles.

4. Prove that the median bisecting the base of an isosceles triangle bisect the vertical angle and is perpendicular to the base.

### 4. Prove that the median bisecting the base of an isosceles triangle bisect the vertical angle and is perpendicular to the base.

3. If \Delta \mathrm{ABC} \cong \Delta \mathrm{LMN} then find the unknown x .

### 3. If \Delta \mathrm{ABC} \cong \Delta \mathrm{LMN} then find the unknown x .

2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(iii) \mathrm{m} \angle \mathrm{A} \cong \ldots \ldots \ldots \ldots

### 2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(iii) \mathrm{m} \angle \mathrm{A} \cong \ldots \ldots \ldots \ldots

2. Identify true and false statement in the following:(i) The sum of the measure of all angles in an rilateral is 360^{\circ} .

### 2. Identify true and false statement in the following:(i) The sum of the measure of all angles in an rilateral is 360^{\circ} .

2. From a point on the bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

### 2. From a point on the bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

1. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=35^{\circ} and m \angle \mathrm{B}=100^{\circ} \overline{\mathrm{BD}} \perp \overline{\mathrm{AC}} . Prove that \triangle B D C is an isosceles triangle.

### 1. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=35^{\circ} and m \angle \mathrm{B}=100^{\circ} \overline{\mathrm{BD}} \perp \overline{\mathrm{AC}} . Prove that \triangle B D C is an isosceles triangle.

Example 2If the bisector of an angle of a triangle bisects the side opposite to it the triangle is isosceles.

### Example 2If the bisector of an angle of a triangle bisects the side opposite to it the triangle is isosceles.

2. Identify true and false statement in the following:(iv) In ân îsosceles triangle corresponding angles and correspoinding sides are equal in measure.

### 2. Identify true and false statement in the following:(iv) In ân îsosceles triangle corresponding angles and correspoinding sides are equal in measure.

3. In the given figure we have \triangle A C E \cong \triangle B D E such that: m \overline{A C}=m \overline{B D}=3 \mathrm{~cm} \angle \mathrm{A}=(3 x+1)^{\circ} m \angle \mathrm{E}= (3 y-2)^{\circ} and m \angle \mathrm{B}=(x+35)^{\circ} . Find the values of x and y .

### 3. In the given figure we have \triangle A C E \cong \triangle B D E such that: m \overline{A C}=m \overline{B D}=3 \mathrm{~cm} \angle \mathrm{A}=(3 x+1)^{\circ} m \angle \mathrm{E}= (3 y-2)^{\circ} and m \angle \mathrm{B}=(x+35)^{\circ} . Find the values of x and y .

3. Fill in the blanks to make the sentences true sentences:(i) In \triangle \mathrm{ABC} \leftrightarrow \triangle \mathrm{DEF} then \overline{\mathrm{AC}} corresponds to

### 3. Fill in the blanks to make the sentences true sentences:(i) In \triangle \mathrm{ABC} \leftrightarrow \triangle \mathrm{DEF} then \overline{\mathrm{AC}} corresponds to

5. Prove that if three altitudes of a triangle are congruent then the triangle is equilatera

### 5. Prove that if three altitudes of a triangle are congruent then the triangle is equilatera

4. Encircle the corresponding letters a b c or d for correct answer:(i) Which of the following is not a sufficent condition for congurence of two triangles?(a) A.S.A\congA.S.A(b) H.S.H \cong H.S.H(c) S.A.A\cong S.A.A(d) A. A.A\cong A.A.A

### 4. Encircle the corresponding letters a b c or d for correct answer:(i) Which of the following is not a sufficent condition for congurence of two triangles?(a) A.S.A\congA.S.A(b) H.S.H \cong H.S.H(c) S.A.A\cong S.A.A(d) A. A.A\cong A.A.A

Example 1If one angle of a right triangled triangle is of 30^{\circ} the hypotenuse is twice as long as the side opposite to the angle.

### Example 1If one angle of a right triangled triangle is of 30^{\circ} the hypotenuse is twice as long as the side opposite to the angle.

1. Prove that any two medians of an equilateral triangle are equal in measure.

### 1. Prove that any two medians of an equilateral triangle are equal in measure.

2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(i) \mathrm{m} \angle \mathrm{M} \cong

### 2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(i) \mathrm{m} \angle \mathrm{M} \cong

EXERCISE 10.4 1. In \triangle \mathrm{PAB} of figure \overline{\mathrm{PQ}} \perp \overline{\mathrm{AB}} and \overline{\mathrm{PA}} \cong \overline{\mathrm{PB}} prove that \overline{\mathrm{AQ}} \cong \overline{\mathrm{BQ}} and \angle \mathrm{APQ} \cong \angle \mathrm{BPQ} .

### EXERCISE 10.4 1. In \triangle \mathrm{PAB} of figure \overline{\mathrm{PQ}} \perp \overline{\mathrm{AB}} and \overline{\mathrm{PA}} \cong \overline{\mathrm{PB}} prove that \overline{\mathrm{AQ}} \cong \overline{\mathrm{BQ}} and \angle \mathrm{APQ} \cong \angle \mathrm{BPQ} .

1. In the given figure \overline{\mathrm{AB}} \cong \overline{\mathrm{CB}} \angle 1 \cong \angle 2 . Prove that \triangle \mathrm{ABD} \cong \triangle \mathrm{CBE} .