#### Chapter 10 Congruent Triangles R.D. Sharma Solutions for Class 9th Exercise 10.1

**Exercise 10.1**

1.Â In Fig. 10.22, the sides BA and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.

**Solution**

2. In a Î”PQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.

**Â Solution**

3. In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that

(i) PT = QT

(ii) âˆ TQR = 15Â°

**Solution**

(ii)

**Solution**

5. In a Î” ABC, if âˆ A = 120Â° and AB = AC. Find âˆ B and âˆ C.

**Solution**Â

**Solution**

7. The vertical angle of an isosceles triangle is 100Â°. Find its base angles.

**Solution**

8. In Fig. 10.24, AB = AC and âˆ ACD = 105Â°, find âˆ BAC.

**Â Solution**

9. Find the measure of each exterior angle of an equilateral triangle.

**Solution**

10. It is given that the base of an isosceles triangle is produced on both sides.

**Solution**

11. In Fig. 10.25, AB = AC and DB = DC, find the ratio âˆ ABD : âˆ ACD.

**Solution**

12. Determine the measure of each of the equal angles of a right-angled isosceles triangle.

OR ABC is a right-angled triangle in which âˆ A = 90Â° and AB = AC. Find âˆ B and âˆ C.

**Solution**

It is given that,

ABC is right angled triangle where,

ABC is right angled triangle where,

âˆ A = 90Â°

AB = ACÂ

We have to find âˆ B and âˆ C.

Since, AB = AC

âˆ B = âˆ CÂ (Isoceles triangle)

Now,

âˆ A +âˆ B+âˆ C = 180Â (Property of triangle)

â‡’ 90 + 2âˆ B = 180Â (âˆ B = âˆ C)

â‡’ 2âˆ B = 180 - 90

â‡’ âˆ B = 45

So,Â âˆ B = âˆ C = 45

Hence,

âˆ B = 45Â° and âˆ C = 45Â°

13. AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.

**Solution**