Question

In adjacent figure AngleABC is isosceles with AB = AC, M is the mid point of BC.
(i) Prove that AngleABM=AngleACM
(ii) Find angleCAM
(iii) Find angleABM
(iv) Find angleAMC
(Hint: Use angle sum property of triangle)
✨Please answer briefly✨ ​

Answer 1

Step-by-step explanation:

(I) To prove : triangle AMB. congruent triangle AMC

AB=AC (Given)

AM=AM(Common)

BM=CM(Given)

so, angleABM=angleABC (CPCT)

(II) Angle BAM=Angle CAM (CPCT)

25 = 25

Angle CAM = 25

(III) Angle ABM =Angle ACM

65=65

Angle ABM=65

(iv) angle CAM + Angle ACM +Angle AMC =180 (Angle sum property of triangle)

25+65+AngleACM=180

90 +Angle ACM =180

Angle ACM =180 -90

Angle ACM =90

Answer 2

SteP-By-StEp ExpLanATiOn:

.

(i) Prove that ∠ABM = ∠ACM.

Given ; ∆ ABC is Isosceles ∆>

• AB = AC
• M is Midpoint of BC.

To Proof ; ∠ABM = ∠ACM

Proof,

In ∆ ABM and ∆ACM

• AB = AC •••[given]
• BM = MC •••[As M is Midpoint of BC]
• AM = AM •••[common]

By SSS congruecy rule,

• ∆ ABM ≅ ∆ACM

So,

• ∠ABM = ∠ACM •••[CPCT]

Hence Proved

(ii) Find Angle CAM.

∠ABC + ∠BCA + ∠CAB = 180° •••[ASPO∆]

• ➤ ∠ABM + ∠ACM + ∠CAB = 180°

As ∠ABM = ∠ACM

• ➤ 65° + 65° + ∠CAB = 180°
• ➤ 130° + ∠CAB = 180°
• ➤ ∠CAB = 50°

∠CAM + ∠BAM = ∠CAB •••[From the figure]

• ➤ ∠CAM + ∠BAM = 50°
• ➤ ∠CAM + 25° = 50°
• ➤ ∠CAM = 25°

So,

• Angle CAM is 25°.

(iii) Find Angle ABM.

∠ABM = ∠ACM

• ➤ ∠ABM = 65°

So,

• Angle ABM is 65°.

(iv) Find Angle AMC.

∠AMC + ∠ACM + ∠CAM = 180° •••[ASPO∆]

• ➤ ∠AMC + 65° + 25° = 180°
• ➤ ∠AMC + 90° = 180°
• ➤ ∠AMC = 90°

So,

• Angle AMC is 90°.