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Maths 3 Mark Question

Congruent Triangles · Maharashtra Board STD 9 (MSBSHSE - English Medium). 5 questions.

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Question 13 Marks
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
Answer
ED is a straight line segment and B and C an points on it.$\angle\text{EBC}=\angle\text{BCD}=\text{straight angle}=180^\circ$
$\angle\text{EBA}+\angle\text{ABC}=\angle\text{ACB}+\angle\text{ACD}$
$\angle\text{EBA}+\angle\text{ACD}+\angle\text{ACB}-\angle\text{ABC}$
$\angle\text{EBA}=\angle\text{ACD}$ [From (i) ABC = ACD]
$\angle\text{ABE}=\angle\text{ACD}$
Hence proved.
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Question 23 Marks
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Answer
The given information and corresponding figure is given below
AB = AD
BC = CD

From the figure, we have
AB = AD (given)
BC = CD (given)
And,
AC = AC (common sides)
Hence, triangles ABC and ADC are congruent to each other.
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Question 43 Marks
In Fig. $\text{AD}\perp\text{CD}$ and $\text{CB}\perp\text{CD}.$ If $\text{AQ}=\text{BP}$ and $\text{DP}=\text{CQ},$ prove that $\angle\text{DAQ}=\angle\text{CBP}.$
Answer
In $\triangle\text{DAQ}$ and $\triangle\text{CBP}$$\angle\text{ADQ}=\angle\text{BCP}=90^\circ$
$\text{DP}=\text{CQ}$ [given]
$\Rightarrow\text{DP}+\text{PQ}=\text{CQ}+\text{PQ}$
$\Rightarrow\text{DQ}=\text{CP}$
$\text{AQ}=\text{BP}$ [given]
By RHS congurence criterion $\triangle\text{DAQ}\cong\triangle\text{CBP}$$\therefore\angle\text{DAQ}=\angle\text{CBP}$ [c.p.c.t]
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Question 53 Marks
If ABC and DEF are two triangles such that AC = 2.5cm, BC = 5cm, $\angle\text{C}=75^\circ,$ DE = 2.5cm DF = 5cm and $\angle\text{D}=75^\circ.$ Are two triangles congruent?
Answer
It is given that $\text{AC}=2.5$
$\text{BC}=5$
$\angle\text{C}=75^\circ$
$\text{DE}=2.5$
$\text{DF}=5$
$\angle\text{D}=75^\circ$
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
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