AAA Similarity

Q. ABCD is a trapezium such that AB||DC. If P and Q are the midpoint of diagonal AC and BD respectively, then show that PQ is equal to half the difference of AB and CD.

Q. Question 8
If in two right triangles, one of the acute angles of one triangle is equal to an acute of the other triangle. Can you say that two triangles will be similar? Why?

Q. 28. In a trapezium ABCD, AB || DC and DC = 2AB. EF || AB, where E and F lie on BC and AD respectively such that BE/EC = 4/3. Diagonal DB intersects EF at G. Prove that, 7EF = 11AB.

Q.

In a $△ABC$, Let P and Q be points on AB and AC respectively such that PQ || BC. Then median AD bisects PQ.

1. True

2. False

Q.

In figure, ABCD is a trapezium with AB||DC. If $△$AED is similar to $△$BEC, which of the following is true?

1. EA = EB
2. EA = AD
3. AD = BC
4. BC = EB

Q. In the given figure, $\square$ABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove $\square$APCQ is a parallelogram.

Q. 39.In a trapezium ABCD, AB is parallel to DC and DC = 3AB. EF drawn parallel to AB which cuts AD at F and BC at E such that AF/FD = 3/5. Diagonal DB intersects EF at G. Find the ratio of EF to AB.

Q.

In the figure given below AB, CD, and EF are parallel lines. Given AB = 9 cm, DC = y cm, EF = 4.5 cm, BC = x cm, and CE = 3 cm. The value of 'x' and 'y' are:

1. 6 cm and 5 cm respectively

2. 3 cm and 6 cm respectively

3. 4.5 cm and 3 cm respectively

4. 6 cm and 3 cm respectively

Q.

$\text{In the given figure, ABCD is a quadrilateral in which}AB||DC\text{and P is the midpoint of BC.}\text{On producing, AP and DC meet at Q. Prove that (i) AB = CQ, (ii) DQ = DC + AB.}$

Q.

In the given figure, DE || BC. If AD = 6 cm, AB = 24 cm and DE = 5 cm, then BC = ____ cm.

1. 20

2. 24

3. 5

4. 10

Q. Observe the given construction, AF is the angular bisector of ∠BAE and AC is the angular bisector of ∠ BAF. If ∠BAE = 60°, then find the value of ∠BAC.

Q. Question 1 (vi)
State whether the pair of triangles in the below figure is similar. Write the similarity criterion used by you for answering the question and also write the pair of similar triangles in the symbolic form:

Q. $\text{In the below figure, if}△ABC\cong △XYZ,\text{then find the}\text{value of x is.}$

Q.

$\text{In the following figure, triangle AXY}\text{is isosceles with}\angle AXY=\angle AYX.$

$\text{If}\frac{BX}{AX}=\frac{CY}{AY},\text{then}$

1. triangle ABC is scalene
2. triangle ABC is isosceles
3. triangle ABC is equilateral
4. None of the above

Q.

In the figure, $△ABC$ is a right angled triangle with $\angle B={90}^{\circ }$. If BD is perpendicular to AC, then $AC\times AD$ = ____.

1. $A{D}^2$
2. $A{B}^2$
3. $B{D}^2$
4. $D{C}^2$

Q. In the adjacent Figure ,

if seg AB || seg PQ , seg AB $\cong$seg PQ, seg AC || seg PR, seg AC $\cong$ seg PR then prove that, seg BC || seg QR and seg BC$\cong$seg QR.

Q.

In the figure given below, AB||EF||CD, If AB =22.5 cm, EP =7.5 cm, PC =15 cm and DC =27 cm.

Calculate: (i) EF (ii) AC

1. AC = 35.5 cm

2. AC = 37.5 cm

3. EF = 13.5 cm

4. EF = 14 cm

Q. the altitude of a triangle are 12; 15 ; 20 . find the largest angle in degree

Q. Question 6
A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

Q.

In $\Delta ABC$, point D and E lies on the line AB and AC respectively as shown in the figure. Find the measure of $\angle AED$.

Q. Minimum number of pairs of equal corresponding angles to make two triangles similar = .

1. 2
2. 3
3. 1

Q.

A vertical stick 20 cm long casts a shadow 10 m long on the ground. At the same time, a tower casts a shadow 50 m long on the ground. The height of the tower is:

1. 100 m

2. 120 m

3. 25 m

4. 200 m

Q. In a triangle ABC, AB = AC then angle B = ______.

1. A
2. C
3. 180 deg - C
4. 180 deg - A

Q.

If triangle ABC is an isosceles triangle in which AB = AC = 13 cm then find the value of $\frac{\text{area of}\Delta ADC}{\text{area of}\Delta EFB}.$

1. $\frac{165}{320}$

2. $\frac{13}{18}$

3. $\frac{150}{319}$

4. $\frac{169}{324}$

Q. In the given figure, ABCD is a parallelogram, O is any point on AC, PQ||AB and LM||AD. Prove that: ar(||gm DLOP)=ar(||gm BMOQ).

Q.

In the given figure, $AB\parallel DC$, BO = 6 cm and DO = 8 cm. Find $BP\times DO$.

Q. 3. ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that Bp×DQ=AB×BC.

Q. In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that $\square$PQRS is a parallelogram.

Q. In the given figure, line PS is a transversal of parallel line AB and line CD. If Ray QX, ray QY, ray RX, ray RY are angle bisectors, then prove that $\boxed{123}$ QXRY is a rectangle.

Q. The diagonals of trapezium PQRS where PQ||RS intersect at point O are given in the figure. Length of OP and OQ is 5 and 4 units, respectively. Find the length of OS, if the length of OR is 2 units.