What additional information is needed to prove the triangles are congruent by SAS postulate?

Congruent triangles are triangles with identical sides and angles. The three sides of one are exactly equal in measure to the three sides of another. The three angles of one are each the same angle as the other.

Triangle Congruence Postulates

Five ways are available for finding two triangles congruent:

1. SSS, or Side Side Side
2. SAS, or Side Angle Side
3. ASA, or Angle Side Side
4. AAS, or Angle Angle Side
5. HL, or Hypotenuse Leg, for right triangles only

Included Parts

An included angle lies between two named sides. In △CAT below, included ∠A is between sides t and c:

An included side lies between two named angles of the triangle.

Side Side Side Postulate

A postulate is a statement taken to be true without proof. The SSS Postulate tells us,

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

Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks.

If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS:

Side Angle Side Postulate

The SAS Postulate tells us,

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

△HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent.

A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent.

Angle Side Angle Postulate

This postulate says,

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

We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.

Angle Angle Side Theorem

We are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,

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

Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters:

[construct as described]

By the AAS Theorem, these two triangles are congruent.

HL Postulate

Exclusively for right triangles, the HL Postulate tells us,

Two right triangles that have a congruent hypotenuse and a corresponding congruent leg are congruent.

The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles.

Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent:

[insert congruent right triangles left-facing △COW and right facing △PIG]

So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions.

Proof Using Congruence

Given: △MAG and △ICG

MC ≅ AI

AG ≅ GI

Prove: △MAG ≅ △ICG

Statement Reason

MC ≅ AI Given

AG ≅ GI

∠MGA ≅ ∠ IGC Vertical Angles are Congruent

△MAG ≅ △ICG Side Angle Side

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

Next Lesson:

Triangle Congruence Theorems

Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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

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