Properties of Similar Triangles

Q.

The side BC of a $△$ABC produced, such that D is on ray BC. The bisector of $\angle$A meets BC in L. Prove that $\angle$ABC + $\angle$ACD = 2$\angle$ALC. [3 MARKS]

Q.

The triangles ABC and ADE are similar.

Which of the following is true ?

1. $\frac{AB}{AD}=\frac{BC}{DE}$
2. $\frac{BC}{BD}=\frac{CE}{DE}$

3. $\frac{EC}{AC}=\frac{AD}{DE}$

4. All of the above.

Q. If two triangles are congruent then the measure of two corresponding sides are __.

1. equal
2. distinct
3. 5 cm
4. unequal

Q. ABC is a triangle and M & N are points on AB and AC such that MN || BC. Find the given ratios if AM:MB = 2:3.

1. $2:3$
2. $2:5$
3. $3:5$

Q. In the given figure,
$BX=\frac{1}{2}AB,BY=\frac{1}{2}BCandAB=AC$.
Which of the following is true?

1. AX = AC
2. BY = AB
3. BX = BY
4. BX = BC

Q. As shown in the following figure, in $∆$ LMN and $∆$PNM, LM = PN, LN = PM. Write the test which assures the congruence of the two triangles. Write their remaining congruent parts.

Q. According to AAA criterion, if the corresponding angles in two triangles are equal, then the triangles are .

1. Symmetrical
2. Equal
3. Similar
4. Congruent

Q. In the given figure, find the perimeter of $△ABC$.

1. 42.5 cm
2. 42 cm
3. 32.5 cm
4. 40 cm

Q.

$△$ ABC is such that AB = 3 cm, BC = 2 cm and CA = 2.5 cm.$△$ DEF is similar to $△$ABC. If EF = 4 cm, then the perimeter of $△$DEF in cm is equal to ___

Q. In the given figure, $△ABC$ is a right-angled triangle and BD is a perpendicular to AC. Which of the following is true?

1. $△ABD\sim △BDC$
2. $△ABD\sim △ABC$
3. $△BCD\sim △ABC$
4. $△BDC\sim △ABC$

Q. Assuming that the two triangles ABC and PQR are similar, is the value of QR if AB=10, PQ=5 and BC=8.

1. 4
2. 7
3. 2

Q. In fig., if PQ $\parallel$ BC and PR $\parallel$CD, then, $\frac{AR}{AD}$=

1. $\frac{AP}{AC}$
2. $\frac{AC}{AP}$
3. None of the above
4. $\frac{AQ}{AB}$

Q. In $△PQR\sim △XYZ,PQ=5,QR=4,PR=7$. Then using SSS criteria, $XY$ and $YZ$ are in the ratio

1. $4:5$
2. $5:4$
3. $4:7$
4. $7:4$

Q. In fig., If PQ $\parallel$ BC and PR $\parallel$CD. Then, $\frac{AR}{AD}$= $\frac{AQ}{AB}$.

1. False
2. True

Q. In $\Delta ABC,AB=BC=6cm$. If a circle is drawn with center at B and radius 2 cm which intersects AB and BC at E and D, then $\Delta ABC\sim \Delta EBD$.

1. True
2. False

Q. Assertion (A): If one angle of a triangle is equal to one angle of another triangle and bisectors of these angles divide the opposite sides in the same ratio, then the triangles are similar.
Reason (R): The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Which of the following is true?

1. Both (A) and (R) are false
2. (A) and (R) are true and (R) is the correct explanation of (A)
3. (A) and (R) are true but (R) is not the correct explanation of (A)
4. (A) is true and (R) is false

Q.

If PQRS is a parallelogram, with point A on QR, then $\frac{AP}{AB}$ is equal to

1. 1

2. 2

3. $\frac{1}{2}$

4. can't be determined

Q. Arrange the sides in decreasing order, if ABC and PQR are similar triangles with AB = 20, PQ = 4 and BC =10. Highest value at the top

1. PQ
2. AC
3. QR
4. PR

Q. In the given figure, ABCD is a quadrilateral with $\angle 1=\angle 2$ and $\angle 3=\angle ABC$ and $\angle 4=\angle BAD$. Also, AP and BQ are produced to points E and F, respectively. Then, we can say that

1. $\Delta OFE$ is an isosceles triangle
2. $\Delta OFE$ is an equilateral triangle
3. $\Delta OFE$ is a scalene triangle
4. Can't be determined

Q. In the given figure if 𝑩𝑬 = 𝑪𝑭, then which of the following option is correct?​

1. 
   △AEB ≅ △ABC
2. 
   △ABC ≅ △ACF
3. 
   △ABE ≅ △CBF
4. 
   △ABC ≅ △ACF

Q. A boy of height 60 cm is walking away from the base of a building at a speed of 2.2 m/s. If the height of the building is 7.2 m, then find the length of his shadow after 5 seconds.

1. $2m$
2. $3m$
3. $1m$
4. $4m$

Q.

In $\Delta ABC$, with points P and Q on sides AB and AC, respectively such that $\frac{AP}{PB}=\frac{AQ}{QC}$. If PQ is extended to T such that PT = BC and PB = TC, then x is equal to

1. ${75}^{\circ }$

2. ${60}^{\circ }$

3. ${120}^{\circ }$

4. ${45}^{\circ }$

Q. In a triangle ABC, right-angled at B, BD is drawn perpendicular to AC. hence, $\angle$CDB = $\angle$A.

1. True
2. False

Q.

The ratio of the corresponding sides of two similar triangles is 1 : 3. The ratio of their corresponding heights is _________.

1. 3:1

2. 1:3

3. 1:9

4. 9:1

Q. The triangles ABC and FEG are similar.
The below are the pairs of corresponding sides :

1. AB and EF
2. AC and FG
3. BC and EG

Which of the following is true ?

1. $\frac{AB}{EF}=\frac{AC}{FG}$
2. All of these
3. $\frac{CD}{AB}=\frac{GH}{HF}$
4. $\frac{BD}{FH}=\frac{BC}{FG}$

Q. In parallelogram $ABCD,E$ is a mid-point of side $AB$ and $CE$ bisect angle $BCD.$ Prove that
Angle $DEC$ is a right angle.

Q. In the given figure, if AB || CD, CD || EF and ∠𝑦 : ∠𝑧 = 4 : 11, find 𝑥.