Question

In the figure, point B is the midpoint of AC . Use the figure to answer the questions.

(a) Jeremy says that ABD= CBD by the SAS congruence postulate. Do you agree or disagree? Explain.
(b) Suppose it is also known that AD=CD . Which postulate or theorem can be used to prove that ABD=CBD ? Justify your answer.

Answer 1

Answer

Proof of (a)

Disagree with the  ΔABD = ΔCBD by the SAS congruence postulate .

SAS congurence property

In this congurence property two sides and one angle of the two triangles are equal .

(1) BD = BD ( Common sides of the  ΔABD and ΔCBD )

(2) AB =BC (B is the midpoint of the AC i.e it bisect AC in the two equal parts.)

but equal angles are not given

therefore disagree with the  ΔABD = ΔCBD  by the SAS congurence postulate .

Proof of (b)

SSS congurence property

In this congurence property three sides of the two triangles are equal .

In the ΔABD and ΔCBD .

(1) BD = BD ( Common sides of the  ΔABD and ΔCBD )

(2) AB =BC (B is the midpoint of the AC i.e it bisect AC in the two equal parts.)

(3) AD=CD  (Given )

ΔABD = ΔCBD  by the SSS congurence theorem

Hence proved

Answer 2

A) Jeremy's statement that ∠ABD ≅ ∠CBD by the SAS congruence postulate; is not true and I will disagree with it because no included angle is given.

B) The postulate that can be used to prove that ∠ABD ≅ ∠CBD if AD = CD is; SSS Congruence postulate.

A) We are given that;

Point B is the midpoint of AC. Thus;

AB = BC

Also, by reflexive property of congruence,

BD = BD

Now, since BD bisects AC into 2 equal parts, but we are not told if the bisector is perpendicular and as such we can't say for sure that ∠ABD = ∠CBD.

Therefore, we have 2 corresponding sides but no angle has been given to be congruent and as such we will disagree with the SAS Congruence postulate used.

B) We are told that AB = CD  

From part A above, we established that AB = BC and BD = BD

Since the three corresponding sides of ΔABD and ΔCBD are congruent to each other, then we can say that ΔABD and ΔCBD are congruent by the SSS Congruence postulate.

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