**(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 **t****riangles 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.*