Question

If Line segment C B. bisects ∠ACD, what additional information could be used to prove ΔABC ≅ ΔDBC using SAS? Select three options.

m∠ABC = 125° and AB ≅ DB
ΔACD is isosceles with base AD
ΔABD is isosceles with base AD
CD = 52 cm
AB = 29 cm

Answer 1

Answer:

Option (1)

Step-by-step explanation:

In the figure attached,

BC is the angle bisector of angle ACD.

To prove ΔABC and ΔDBC congruent by SAS property we require two sides and the angle between these sides to be congruent.

Since BC ≅ BC [Reflexive property]

∠ABC ≅ ∠CBD ≅ 125°

And sides AB ≅ BD

Both the triangles will be congruent.

Therefore, additional information required to prove ΔABC ≅ ΔDBC have been given in option (1).

Therefore, Option (1) will be the answer.

Answer 2

The additional information that could be used to prove ΔABC ≅ ΔDBC

using SAS are;

1. m∠ABC = 125° and $\overline{AB}$ ≅ $\overline{DB}$
2. ΔACD is isosceles, with base $\overline{AD}$
3. $\overline{CD}$ = 52 cm

Reasons:

The given information are;

$\overline{CB}$ bisects ∠ACD

The given information from the diagram are;

$\overline{AC}$ = 52 cm

$\overline{BD}$ = 29 cm

∠CBD = 125°

Solution;

First selected option; m∠ABC = 125° and $\overline{AB}$ ≅ $\overline{DB}$

• m∠ABC = 125° and $\overline{AB}$ ≅ $\overline{DB}$

$\overline{CB}$ ≅ $\overline{CB}$ (reflexive property)

ΔABC ≅ ΔDBC (SAS rule of congruency)

Second selected option; ΔACD is isosceles, with base $\overline{AD}$

∠ACB ≅ ∠DCB (definition of angle bisector)

• ΔACD is isosceles, with base $\overline{AD}$ (Additional information)

$\overline{CD}$ ≅ $\overline{AC}$ (definition of isosceles triangle)

$\overline{CB}$ ≅ $\overline{CB}$ (reflexive property)

ΔABC ≅ ΔDBC (SAS rule of congruency)

Third selected option;  $\overline{CD}$ = 52 cm

∠ACB ≅ ∠DCB (definition of angle bisector)

• $\overline{CD}$ = 52 cm = $\overline{AC}$ (given) (additional information)

$\overline{CD}$ ≅ $\overline{AC}$ (definition of congruency)

$\overline{CB}$ ≅ $\overline{CB}$ (reflexive property)

ΔABC ≅ ΔDBC (SAS rule of congruency)

The Side-Angle-Side, SAS, rule of congruency states that two triangles are

congruent if two sides and an included angle of one triangle are

congruent to the corresponding two sides and included angle on the other

triangle.

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