True False[1 Marks ]

Question 11 Mark

If the bisector of the verical angle of a triangle bisects the base, then the triangle may be isosceles.

Answer

The angular bisector of the vertex angle is also a mediam.
$\Rightarrow$ The tirnagle must be a isosceles and also may be an equilateral triangle.

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Question 21 Mark

An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Answer

According to exterior angle theorem, $\text{ext.x}=\angle\text{CAB}+\angle\text{CBA}$

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Question 31 Mark

An exterior angle of a triangle is greater than the opposite interior angles.

Answer

According to exterior angle theorem,
$\text{ext.x}=\angle\text{CAB}+\angle\text{CBA}$
Since, the exterior angle is the sum of its interior angels.
Thus,
$\text{ext.x}>\angle\text{CAB}$
$\text{ext.x}>\angle\text{CBA}$

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Question 41 Mark

Sum of the three angles of a triangle is $180^\circ .$

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$

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Question 51 Mark

All the angles of a triangle can be equal to $60^\circ .$

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$
$\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
Now, if all the three angles of a triangle are equal to $60^\circ$
Then, $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$

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Question 61 Mark

Angles opposite to equal sides of a triangle are equal.

Answer

True. Explenation: Since the sides are equal, the corresponding opposite angles must be equal.

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Question 71 Mark

Sum of any two sides of a triangle is greater than twice the median drown to the third side.

Answer

Given: In triangle $ABC, AD$ is the median drawn from $A$ to $BC.$
To prove: $AB + AC > AD$ Construction: Produce $AD$ to $E$ so that $DE = AD,$ Join $BE.$
Proof: Now in $\triangle\text{ADC}$ and $\triangle\text{EDB},$
$AD = DE ($by const$) DC = BD($as $D$ is mid-point$)$
$\angle\text{ADC}=\angle\text{EDB}$ $\big($vertically opp, $\angle\text{S}\big)$
Therefore, $\triangle\text{ABE}, \triangle\text{ADC}\cong\triangle\text{EDB} ($by $SAS)$
This gives, $BE = AC. AB + BE > AE AB + AC > 2AD$
$\big(\therefore AD = DE$ and $BE = AC \big)$
Hence the sum of any two sides of a triangle is greater than the median drawn to the third side.

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Question 81 Mark

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.

Answer

True. Explanation: The perpendicular distance is the shortest distance from a point to a line not containing it.

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Question 91 Mark

The two alitutde corresponding to two equal sides of a triangle need not be equal.

Answer

False.Explenation:
Since two sides are equal, the triangle is an isosceles triangle. The two altitudes corresponding to two equal side must be equal.

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Question 101 Mark

The bisectors of two equal angles of a triangle are equal.

Answer

True.Explenation:
Since it an isosceles triangle, the lenghts of bisectors of the two equal angles are equal.

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Question 111 Mark

A triangle can have two right angles.

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
Now, if there are two right angles in the triangle Let $\angle\text{B}=\angle\text{C}=90^\circ$
Then, $\angle\text{A}+90^\circ+90^\circ=180^\circ$
$\angle\text{A}+180^\circ=180^\circ$
$\angle\text{A}=180^\circ-180^\circ$
$\angle\text{A}=0^\circ$ (This is not possible.)

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Question 121 Mark

Sum of the three sides of a triangle is less than the sum of its three altitudes.

Answer

False. Explanation: Sum of these sides of a triangle is greater than sum of its three altitudes.

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Question 131 Mark

All the angles of a triangle can be less than $60^\circ .$

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
Now, If all the three angles of a triangle is less than $60^\circ $
Then, $\angle\text{A}+\angle\text{B}+\angle\text{C}<180^\circ$

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Question 141 Mark

If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.

Answer

True. Explanation: The side opposite to greater angle is longer and smaller angle is shorter in a triangle.

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Question 151 Mark

If one angle of a triangle is obtuse, then it cannot be a right angled triangle.

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
Now, if a right angled triangle
Then, $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
$90^\circ+\angle\text{B}+\angle\text{C}=180^\circ$
$\angle\text{B}+\angle\text{C}=90^\circ$
Also if one of the angle's is obtuse $\angle\text{B}+\angle\text{C}>90^\circ$
This is not possible. Thus, if one angle of a triangle is obtuse,
then it cannot be a right angled triangle.

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Question 161 Mark

Side opposite to equal angles of a triangle may be unequal.

Answer

False. Explenation: Side opposite to equal angles of a triangle are equal.

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Question 171 Mark

An exterior angle of a triangle is less than either of its interior opposite angles.

Answer

According to the exterior angle property, an exterior angle of a equal to the sum of the two opposite interior angles.
In $\triangle\text{ABC}$
Let $x$ be the exterior angle
So, $\text{x}=\angle\text{CAB}+\angle\text{CBA}$
Now, if $x$ is less than either of its interior opposite angles $\text{x}<\angle\text{CAB}+\angle\text{CBA}$

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Question 181 Mark

Difference of any two sides of a triangle is equal to the third side.

Answer

False. Explanation: The difference of any two sides of a triangle is less than third side.

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Question 191 Mark

If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

Answer

False.
Explenation:
Here the altitude from the vertex is also the perpendicular bisector of the opposite side.
The triangle must be isosceles and may be an equilateral triangle.

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Question 201 Mark

Sum of any two sides of a triangle is greater than the third side.

Answer

Given $\triangle\text{ABC},$ extend $BA$ to $D$ such that $AD = AC.$
Now in $\triangle\text{DBC},$
$\angle\text{ADC} = \angle\text{ACD } $ [Angles opposite to equal sides are equal]
Hence $\angle\text{BCD} > \angle\text{BDC}$
That is $BD > BC [$The side opposite to the larger $($greater$)$ angle is longer$]$
$⇒ AB + AD > BC$
$\therefore AB + AC > BC [$Since $AD = AC)$
Thus sum of two sides of a triangle is always greater than third side.

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Question 211 Mark

Two right triangles are congruent if hypoenuse and a side of the other trianlge.

Answer

According to $RHS$ congruence criterion the given statment is true.

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Question 221 Mark

A triangle can have at most one obtuse angles.

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$ Now, if a triangle will have more than one obtuse angle Then, $\angle\text{A}+\angle\text{B}+\angle\text{C}>180^\circ$

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Question 231 Mark

All the angles of a triangle can be greater than $60^\circ .$

Answer

According to the angle sum property of the triangle In $\triangle\text{ABC}$ $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
Now, if all the three angles of a triangle is greater than $60^\circ $
Then, $\angle\text{A}+\angle\text{B}+\angle\text{C}>180^\circ$

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Question 241 Mark

If any two sides of a right triangle are respectively equal to two sides of other right triangle then the two triangle are congruent.

Answer

False.Explenation:
The two right triangles may or may not be congrunt.

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Question 251 Mark

The measure of each angle of an equilateral triangle $60^\circ .$

Answer

Since all the three angles of equilateral triangles are equal and sum of the three angles is $180^\circ ,$ each angle will be equal to $\frac{180^\circ}{3}=60^\circ.$

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