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## NCERT Class 9 Maths Textbook Chapter 7 With Answer Book PDF Free Download

### Chapter 7: Triangles

#### 7.1 Introduction

You have studied triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle.

(‘Tri’ means ‘three’). A triangle has three sides, three angles, and three vertices.

For example, in triangle ABC, denoted as Δ ABC (see Fig. 7.1); AB, BC, CA are the three sides, ∠ A, ∠ B, ∠ C are the three angles, and A, B, C are three vertices.

#### 7.3 Criteria for Congruence of Triangles

In earlier classes, you have learned four criteria for the congruence of triangles. Let us recall them.

#### 7.5 Some More Criteria for Congruence of Triangles

You have seen earlier in this chapter that the equality of three angles of one triangle to three angles of the other is not sufficient for the congruence of the two triangles.

You may wonder whether the equality of three sides of one triangle to three sides of another triangle is enough for the congruence of the two triangles. You have already verified in earlier classes that this is indeed true.

Author | NCERT |

Language | English |

No. of Pages | 26 |

PDF Size | 697 KB |

Category | Mathematics |

Source/ Credits | ncert.nic.in |

### NCERT Solutions Class 9 Maths Chapter 7 Triangles

**1. In quadrilateral ACBD, AC = AD and AB bisect ∠A (see Fig. 7.16). Show that ΔABC≅ ΔABD. What can you say about BC and BD?**

**Solution:**

It is given that AC and AD are equal i.e. AC = AD and the line segment AB bisects ∠A.

We will have to now prove that the two triangles ABC and ABD are similar i.e. **ΔABC ≅ ΔABD**

**Proof:**

Consider the triangles ΔABC and ΔABD,

(i) AC = AD (It is given in the question)

(ii) AB = AB (Common)

(iii) ∠CAB = ∠DAB (Since AB is the bisector of angle A)

So, by **SAS congruency criterion**, ΔABC **≅ **ΔABD.

For the 2^{nd} part of the question, BC and BD are of equal lengths by the rule of C.P.C.T.

**2. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that**

(i) ΔABD **≅ **ΔBAC

(ii) BD = AC

(iii) **∠**ABD = **∠**BAC.

**Solution:**

The given parameters from the questions are **∠**DAB = **∠**CBA and AD = BC.

(i) ΔABD and ΔBAC are similar by SAS congruency as

AB = BA (It is the common arm)

**∠**DAB = **∠**CBA and AD = BC (These are given in the question)

So, triangles ABD and BAC are similar i.e. ΔABD **≅ **ΔBAC. (Hence proved).

(ii) It is now known that ΔABD **≅ **ΔBAC so,

BD = AC (by the rule of CPCT).

(iii) Since ΔABD **≅ **ΔBAC so,

Angles **∠**ABD = **∠**BAC (by the rule of CPCT).

NCERT Class 9 Maths Textbook Chapter 7 With Answer Book PDF Free Download