What other information do you need in order to prove the triangles congruent by SAS congruence?

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule.
In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent.

The SSS rule states that:
If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ.

Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent.

The SAS rule states that:
If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

Included Angle           Non-included angle

For the two triangles below, if AC = PQ, BC = PR and angle C< = angle P, then by the SAS rule, triangle ABC is congruent to triangle QRP.

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The ASA rule states that:
If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The AAS rule states that:
If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle R, then triangle ABC is congruent to triangle QRP.

Three Ways To Prove Triangles Congruent

A video lesson on SAS, ASA and SSS.

1. SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
2. SAS Postulate: If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
3. ASA Postulate: If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

• Show Video Lesson

Using Two Column Proofs To Prove Triangles Congruent

Triangle Congruence by SSS How to Prove Triangles Congruent using the Side Side Side Postulate?

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

• Show Video Lesson

Triangle Congruence by SAS How to Prove Triangles Congruent using the SAS Postulate?

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

• Show Video Lesson

Prove Triangle Congruence with ASA Postulate How to Prove Triangles Congruent using the Angle Side Angle Postulate?

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

• Show Video Lesson

Prove Triangle Congruence by AAS Postulate How to Prove Triangles Congruent using the Angle Angle Side Postulate?

If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

• Show Video Lesson

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

(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.