NCERT solutions for Mathematics [English] Class 10 chapter 6 - Triangles [Latest edition]

All circles are ______.

All squares are ______.

All ______ triangles are similar.

Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______ and (b) their corresponding sides are ______. (equal, proportional)

Give two different example of a pair of similar figures.

Give two different examples of pair of Non-similar figures.

State whether the following quadrilaterals are similar or not:

See the given figure. DE || BC. Find EC.

See the given figure. DE || BC. Find AD.

E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR.

PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR:

PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR.

PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

In the following figure, if LM || CB and LN || CD, prove that `("AM")/("AB")=("AN")/("AD")`

In the following figure, DE || AC and DF || AE. Prove that `("BF")/("FE") = ("BE")/("EC")`

In the following figure, DE || OQ and DF || OR, show that EF || QR.

In the following figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that `("AO")/("BO") = ("CO")/("DO")`

The diagonals of a quadrilateral ABCD intersect each other at the point O such that `("AO")/("BO") = ("CO")/("DO")`. Show that ABCD is a trapezium.

State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

State which pair of triangles in the given figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:

In the following figure, ΔODC ∼ ΔOBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB.

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using similarity criterion for two triangles, show that `"OA"/"OC"="OB"/"OD"`.

In the following figure, `("QR")/("QS") = ("QT")/("PR")` and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.

S and T are point on sides PR and QR of ΔPQR such that ∠P = ∠RTS. Show that ΔRPQ ∼ ΔRTS.

In the following figure, if ΔABE ≅ ΔACD, show that ΔADE ∼ ΔABC.

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔAEP ∼ ΔCDP

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔABD ∼ ΔCBE

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔAEP ∼ ΔADB

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔPDC ∼ ΔBEC

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB.

In the following figure, ABC and AMP are two right triangles, right-angled at B and M respectively, prove that:

1. ΔABC ~ ΔAMP
2. `("CA")/("PA") = ("BC")/("MP")`

CD and GH are, respectively, the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG, respectively. If ΔABC ~ ΔFEG, Show that

1. `("CD")/("GH") = ("AC")/("FG")`
2. ΔDCB ~ ΔHGE
3. ΔDCA ~ ΔHGF

In the following figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ΔABD ∼ ΔECF.

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR show that ΔABC ~ ΔPQR.

D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that` \frac{"CA"}{"CD"}=\frac{"CB"}{"CA"} or "CA"^2 = "CB" × "CD".`

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ΔABC ~ ΔPQR. 

A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that `("AB")/("PQ") = ("AD")/("PM")`.