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This is one of our many quizzes on Triangle congruence. Click on this to access more such competitive quizzes on Geometry.

Triangle congruence Quiz 2

mathematics

 Consider an isosceles triangle ABC such that the length of the equal side is 5cm and perimeter is 25cm. Another triangle PQR is congruent to ABC. Find the length of the unequal side in the congruent triangle. 

 For congruent triangles the length of corresponding sides is equal. Two sides of the isosceles triangle are equal.  

Perimeter $=$ Sum of all sides $ \\ $ For $\vartriangle ABC$, perimeter $\vartriangle ABC = AB + BC + CA$ $ \\ $ Suppose $CA$ be the unequal side, $25 = 5+5+x$ $ \\ \Rightarrow x=25-10 \\ \Rightarrow {x=15cm} \\ $ Since $\vartriangle ABC\cong \vartriangle PQR$ the sides will be equal $ \\ $  Hence for the congruent triangle, Length $=15cm$ 

 A mechanical engineer wants to ensure that a quadrilateral is exactly symmetrical, for this he cuts the quadrilateral into two triangles. Which of the following conditions is not enough to satisfy that the triangles are same.

 For the triangles to be symmetric, they must be congruent and hence the condition of congruency must be fulfilled.

If All three sides of both triangles are equal the sides are congruent by SSS condition, and hence symmetric. $ \\ $  If the adjacent sides and the angle between them are equal in both triangles and hence congruent by SAS condition. $ \\ $ If the adjacent sides and the angle between them are equal in both triangles and hence congruent by ASA condition. $ \\ $ The adjacent sides and the next angle is equal in both triangles by SSA is not a congruence condition, so its not satisfying condition

 Meera knows that triangles $XYZ$ and $PQR$ are congruent.  If $\angle X={{70}^{\circ }}$ and $\angle Y={{40}^{\circ }}$ Help Meera in finding the value of $\angle R$ ?

 Angle sum property of a triangle states that the sum of interior angles of a triangle is 180 degrees. Find the corresponding angle of congruent triangles and use angle sum property.

Given $\vartriangle XYZ\cong \vartriangle PQR$ $ \\ \therefore \angle X=\angle P=70 \\  \angle Y=\angle Q=40 \\  \angle Z=\angle R \\ $ Angle sum property of triangle states that the sum of interior angles of a triangle is $180^\circ$ $ \\ \Rightarrow \angle P+\angle Q+\angle R={{180}^{\circ }} \\  70+40+\angle R=180 \\  \angle R=180-110=70 $

 Gautam needs your help in finding which rule of congruence can be used to prove that the triangles are congruent in the figure. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a924c05e-b723-4955-8a8e-090468bac32a1291911088609382578.png"/> 

 It is given that one side is equal, one side is common. Also two angles are equal since, both are of 90 degrees.

In $\vartriangle ABC$ and $\vartriangle ADC$ $ \\ $ Given, $AB=AD$ $ \\ \angle ABC=\angle ADC$ $ \\ $ Since both are $90^{\circ }$ $ \\ $ AC is common in both triangles, so by RHS condition $ \\ \vartriangle ABC \cong \vartriangle ADC$

 In the given figure, are the two triangles of equal measurement? If yes by which condition? <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8a092b82-c3bb-4172-8964-c8c11ee7ce293065093197454982276.png"/>

 One side is common hence it will be equal in both the triangles. Also it is given that two angles are equal. 

In $\vartriangle ABC$ and $\vartriangle ADC$ $ \\ $ Given, $\angle BAC=\angle DAC$ $ \\ $ Also, $\angle BCA=\angle DCA$ $ \\ $ Since AC is common in both triangles, so by ASA condition, $ \\ \vartriangle ABC \cong \vartriangle ADC$