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What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate?

(A) $ \angle BAC  \cong  \angle DAC $

(B) $ AC  \cong  \angle BD $

(A) $ \angle BCA  \cong   \angle DCA $

(A) $ AC \cong  BD $

This article aims to prove that triangles are congruent using the SAS congruence postulate. To prove this statement, the reader should know about reflexive property and line segment theorem.

The reflexive property of congruence is stated as:

– If $  \angle  A $ is an angle, then $ \angle A \cong \angle  A $.

– If $ \bar { AB } $ is a line segment, then $ \bar  { AB } \cong  \bar { AB } $.

– If $ O $ is the shape, then $ O \cong O $.

The line segment theorem states that

The points perpendicular to the axis of the line are equidistant from the endpoints of the line is a theorem.

Expert Answer

Step 1

Given: The triangles are

Step 2

Use the SAS congruence postulate to determine what information is needed to prove the congruence of triangles. To verify the SAS congruence postulate, we need to prove that two sides and one angle are congruent in a triangle $ \Delta ACB  $ and $  \Delta ACD  $.

Using the given diagram $ BC $ is congruent $  CD  $ to prove $ \Delta ACB \cong  \Delta ACD $. $  AC  $ is congruent to $ AC $, Using reflective properties.

In triangle $ ABC $, $ AC $ is the bisector of angle $ A $ and the bisector of side $ BD $

Using the line segment theorem

\[ \triangle BAC  \cong  \triangle DAC \]

Therefore, to prove that triangles are congruent using the SAS congruence postulate, you need information $ \triangle BAC \cong DAC $

Numerical Result

To prove that triangles are congruent using the SAS congruence postulate, you need information $\triangle BAC \cong DAC $.

Example

What other information do I need to prove that the triangles are congruent using the SAS Congruence Postulate?

Solution

$ AC $ is perpendicular to $ BD $.

Given a triangle $ ABD $. $ C $ is the midpoint of $ BD $.

We need to use the SAS hypothesis to prove that two triangles are congruent.

Here consider two triangles $ ABC $ and $ ADC $

Reason for statement

1) $ BC = CD $ $ D $ is the midpoint of $ BD $

2) $ AC = AC $ Reflective property

Since we have a congruence of two sides, we must also include an angle congruence

i.e. $ Angle\: ACB  = Angle\:  ACD $

If this information is given, then this completes the SAS congruence for the two triangles $ ABC $ and $ ADC $

So the answer is

The information that $ AC $ is perpendicular to $ BD $ is sufficient to complete the proof.

Images/Mathematical drawings are created with Geogebra.

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