Question

ABC is an isosceles triangle in which AB equal to AC and bisector of exterior angle QACand CD parallel to BA yashovan figure as shown in the figure. show that ​

Answer 1

Step-by-step explanation:

ABC is an isosceles triangle in which AB = AC. AD bisects ∠PAC and CD || AB. Show that

(i) ∠DAC = ∠BCA

(ii) ABCD is a parallelogram

Given: ABC is an isosceles triangle in which AB = AC. AD bisects ∠PAC and CD || AB.

To Prove:

(i) ∠DAC = ∠BCA

(ii) ABCD is a parallelogram.

Proof:

(i) In ∆ABC,

∵ AB = AC

∴ ∠B = ∠C ...(1)

| Angles opposite to equal sides of a triangle are equal

Also, Ext. ∠PAC = ∠B + ∠C

| An exterior angle of a triangle is equal to the sum of two interior opposite angles

⇒ ∠PAC = ∠C + ∠C | From (1)

⇒ 2 ∠CAD = 2 ∠C | ∵ AD bisects ∠PAC

⇒ ∠CAD = ∠C

⇒ ∠DAC = ∠BCA

(ii) But these angles form a pair of equal alternate interior angles

∴ AD || BC

Also, CD || AB

∴ ABCD is a parallelogram

| A quadrilateral is a parallelogram if its both the pairs of opposite sides are parallel.