#### Chapters

## Chapter 3: Triangles

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Practice Set 3.1 [Pages 27 - 28]

In the given figure, ∠ ACD is an exterior angle of Δ ABC . ∠ B = 40^{°}, ∠ A = 70^{°}.

Find the measure of ∠ ACD.

In Δ PQR, ∠ P = 70^{°}, ∠ Q = 65 ° then find ∠ R.

The measures of angles of a triangle are x^{°}, ( x - 20)^{°}, (x - 40)°. Find the measure of each angle.

The measure of one of the angles of a triangle is twice the measure of its smallest angle and the measure of the other is thrice the measure of the smallest angle. Find the measures of the three angles.

In the given figure, measures of some angles are given. Using the measures find the values of x, y, z.

In the given figure, line AB || line DE. Find the measures of ∠ DRE and ∠ ARE using given measures of some angles.

In Δ ABC, bisectors of ∠ A and ∠ B intersect at point O. If ∠ C = 70^{°} . Find measure of ∠ AOB.

In the given Figure, line AB || line CD and line PQ is the transversal. Ray PT and ray QT are bisectors of ∠ BPQ and ∠ PQD respectively.

Prove that m ∠ PTQ = 90 °.

Using the information shown in figure, find the measures of ∠ a, ∠ b and ∠ c.

In the given figure, line DE || line GF ray EG and ray FG are bisectors of ∠ DEF and ∠ DFM respectively. Prove that,

1. `angle DEG = 1/2 angle EDF`

2. EF = FG

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Practice Set 3.2 [Pages 31 - 33]

In the following example , a pair of triangles is shown. Equal parts of triangles in each pair are marked with the same sign. Observe the figure and state the test by which the triangle in each pair are congruent.

By . . . . . . . . . . test

Δ ABC ≅ ΔPQR

In the following example , a pair of triangles is shown. Equal parts of triangle in each pair are marked with the same sign. Observe the figure and state the test by which the triangle in each pair are congruent.

By . . . . . . . . . . test

Δ XYZ ≅ Δ LMN

In the following example , a pair of triangles is shown. Equal parts of triangle in each pair are marked with the same sign. Observe the figure and state the test by which the triangles in each pair are congruent.

By . . . . . . . . . . test

Δ LMN ≅ Δ PTR

In the following example ,a pair of triangles is shown. Equal parts of triangle in each pair are marked with the same sign. Observe the figure and state the test by which the triangle in each pair are congruent.

By . . . . . . . . . . test

Δ LMN ≅ Δ PTR

**}**..correspoding sides of congruent triangles

Observe the information shown in pair of triangle given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.

From the information shown in the figure,

in ΔPTQ and Δ STR

seg PT ≅ seg ST

∠ PTQ ≅ ∠ STR ....Vertically opposite angles

∴ Δ PTQ ≅ Δ STR .... _____ test

∴ ∠ TPQ ≅ _____ and ____ ≅ ∠ TRS .....corresponding angles of congruent triangles

seg PQ ≅ ______ corresponding sides of congruent triangles

From the information shown in the figure, state the test assuring the congruence of Δ ABC and Δ PQR. Write the remaining congruent parts of the triangles.

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

In the given figure, seg AB ≅ seg CB and seg AD ≅ seg CD.

Prove that Δ ABD ≅ Δ CBD

In the given figure, ∠ P ≅ ∠R seg PQ ≅ seg RQ

Prove that, Δ PQT ≅ Δ RQS

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Practice Set 3.3 [Page 38]

Find the values of x and y using the information shown in figure .

Find the measure of ∠ ABD and m∠ ACD.

The length of hypotenuse of a right angled triangle is 15. Find the length of median of its hypotenuse.

In Δ PQR, ∠ Q = 90^{° }, PQ = 12, QR = 5 and QS is a median. Find l(QS).

In the given figure, P point G is the point of concurrence of the medians of Δ PQR . If GT = 2.5, find the lengths of PG and PT.

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Practice Set 3.4 [Pages 43 - 44]

In the given figure, point A is on the bisector of ∠ XYZ. If AX = 2 cm then find AZ.

In the given figure, ∠ RST = 56^{°} , seg PT ⊥ ray ST, seg PR ⊥ ray SR and seg PR ≅ seg PT Find the measure of ∠ RSP. State the reason for your answer.

In Δ PQR, PQ = 10 cm, QR = 12 cm, PR = 8 cm. Find out the greatest and the smallest angle of the triangle.

In Δ FAN, ∠ F = 80^{° }, ∠ A = 40^{° }. Find out the greatest and the smallest side of the triangle. State the reason.

Prove that an equilateral triangle is equiangular.

Prove that, if the bisector of ∠ BAC of Δ ABC is perpendicular to side BC, then Δ ABC is an isosceles triangle.

In the given figure, if seg PR ≅ seg PQ, show that seg PS > seg PQ.

In the given figure, in Δ ABC, seg AD and seg BE are altitudes and AE = BD. Prove that seg AD ≅ seg BE

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Practice Set 3.5 [Page 47]

If Δ XYZ ~ Δ LMN, write the corresponding angles of the two triangles and also write the ratios of corresponding sides.

In Δ XYZ, XY = 4 cm, YZ = 6 cm, XZ = 5 cm, If Δ XYZ ~

Δ PQR and PQ = 8 cm then find the lengths of remaining sides of Δ PQR.

Draw a sketch of a pair of similar triangles. Label them. Show their corresponding angles by the same signs. Show the lengths of corresponding sides by numbers in proportion.

#### Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry Chapter 3 Triangles Exercise Problem Set 3 [Pages 49 - 50]

Choose the correct alternative answer for the following question.

If two sides of a triangle are 5 cm and 1.5 cm, the lenght of its third side cannot be . . . . . . . .

3.7 cm

4.1 cm

3.8 cm

3.4 cm

Choose the correct alternative answer for the following question.

In Δ PQR, If ∠ R > ∠ Q then . . . . . . . .

QR > PR

PQ > PR

PQ < PR

QR < PR

In Δ TPQ, ∠ T = 65^{°} , ∠ P = 95^{°} which of the following is a true statement ?

PQ < TP

PQ < TQ

TQ < TP < PQ

PQ < TP < TQ

Δ ABC is isosceles in which AB = AC. Seg BD and seg CE are medians. Show that BD = CE.

In Δ PQR, If PQ > PR and bisectors of ∠ Q and ∠ R intersect at S. Show that SQ > SR.

In the figure, point D and E are on side BC of Δ ABC, such that BD = CE and AD = AE.

Show that Δ ABD ≅ Δ ACE.

In the given figure, point S is any point on side QR of Δ PQR Prove that : PQ + QR + RP > 2PS

In the given figure, bisector of ∠ BAC intersects side BC at point D. Prove that AB > BD

In the given figure, seg PT is the bisector of ∠ QPR. A line through R intersects ray QP at point S. Prove that PS = PR

In the given figure, seg AD ⊥ seg BC. seg AE is the bisector of ∠ CAB and C - E - D. Prove that

∠DAE = `1/2` (∠C - ∠B)

## Chapter 3: Triangles

## Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry chapter 3 - Triangles

Balbharati solutions for Maharashtra state board Class 9 Mathematics 2 Geometry chapter 3 (Triangles) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Maharashtra state board Class 9 Mathematics 2 Geometry solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Maharashtra state board Class 9 Mathematics 2 Geometry chapter 3 Triangles are Concept of Triangles, Theorem of remote interior angles of a triangle, Congruence of Triangles, Isoscles Triangle Theorem, Property of 30-60-90 Triangle Theorem, Median of a Triangle, Perpendicular bisector Theorem, Angle bisector theorem, Properties of inequalities of sides and angles of a triangle, Similar Triangle.

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