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Home / Geometry / Exercise 10.1|Class seven| Congruence| Part – 2

# Exercise 10.1|Class seven| Congruence| Part – 2

Problem – 5: In the figure, AD = AE, BD = CE and ∠AEC = ∠ADB. Prove that AB = AC.

Particular enunciation: In the figure, AD = AE, BD = CE and ∠AEC = ∠ADB. We have to prove that AB = AC.

BD = CE [Given]

Therefore, AB = AC.  (Proved)

Problem – 6: In the figure, ∆ABC and ∆DBC are both isosceles triangles. Prove that, ∆ABD ≌ ∆ACD.

Particular enunciation: In the figure, ∆ABC and ∆DBC are both isosceles triangles. We have to prove that, ∆ABD ≌ ∆ACD.

Proof: Given that, ∆ABC is an isosceles triangle.

∴ AB = AC

And ∆DBC is an isosceles triangle.

∴ DB = DC

Now, in ∆ABC and ∆ACD

AB =AC

DB = DC

∴ ∆ABD ≌ ∆ACD (Proved)

Problem – 7: Show that the medians drawn from the extremities of the base of an isosceles triangle to the opposite sides are equal to each other.

General enunciation: The medians drawn from the extremities of the base of an isosceles triangle to the opposite sides are equal to each other.

Particular enunciation: Let, ABC is an isosceles triangle where AB = AC. BQ and CP are medians drawn to the sides AC and AB respectively. We have to prove that, BQ = CP.

Proof: In ∆ABC,

AB =AC

⟹ ½ AB = ½ AC [Divide both side by 2]

⟹ BP = CQ [P and Q are the mid points of AB and AC respectively]

Now in ∆BCE and ∆DCB,

BP = CQ

∴ ∠PBC = ∠QCB [∵ opposite angles of equal arms AB and AC are equal]

And BC is the common side.

∴ ∆BCP ≌ ∆QCB

∴ BQ = CP [proved]

Problem – 8: Prove that the angles of an equilateral triangle are equal to one another.

General enunciation: We have to prove that the angles of an equilateral triangle are equal to one another.

Particular enunciation: Let, ABC is an equilateral triangle i.e., AB = AC = BC.

We have to prove that, ∠A = ∠B = ∠C

Proof: Given that, AB =AC = BC

In ∆ABC,

AB = AC

∴ ∠ACB = ∠BAC [∵opposite angles are equal arms are equal.]

⟹∠C = ∠B —————————————- (1)

Again, in ∆ABC,

AC = BC

∴ ∠ABC = ∠BAC [∵opposite angles are equal arms are equal.]

⟹ ∠B = ∠A ————————————— (2)

From (1) and (2) we have

∠C = ∠B = ∠A

∴ ∠A = ∠B = ∠C [proved]

Exercise 10.1|Class seven| Congruence| Part – 1

## If two angles of triangles are equal, then the sides opposite to the equal angles are equal.

If two angles of triangles are equal, then the sides opposite to the equal angles ...