When two sides and the included angle of the first triangle are congruent to the two sides and the included angle of the second triangle?

We have seen how inequalities can be applied to the sides and angles of a single triangle. Now, we will take a look at how inequalities can be put to work between two triangles.

Hinge Thm If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. (May also be referred to as the SAS Inequality Theorem)

The "included angle" is the angle formed by the two sides of the triangle mentioned in this theorem.

This theorem is called the "Hinge Theorem" because it acts on the principle of the two sides described in the triangle as being "hinged" at their common vertex. Consider the alligator jaws at the right.
The sides described in this theorem are the jaw lengths of the alligator with the "hinge" being at the corner of the alligator's mouth (point A or D). While the jaw lengths of the alligator will not change, the jaw "hinge" does allow the alligator to open, or close, its mouth with varying angular degrees (at point A or D).

Hinge Thm Converse: If two sides of a triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second, then the included angle in the first triangle is greater than the included angle in the second triangle. (May also be referred to as the SSS Inequality Theorem.)

If we return to the alligator analogy, the converse of the Hinge Theorem would tell us that the wider the alligator opens his mouth (EF > BC), the larger the angle he creates at the hinge of his jaw (m∠D > m∠B).            If EF > BC, then m∠D > m∠B.

1. Fill the box with >, <, or =. Solution: Since , the conditions for the Hinge Theorem are satisfied, and the longer segment (side) will be opposite the larger angle. Answer: AR < BR

2. Given AC = 18, AD = 18, m∠CAB = 31º, m∠BAD = (2x - 3)º Write an inequality, or set of inequalities, to describe the possible values for x. Solution: AB = AB, so the Converse of the Hinge Theorem applies. Since CB > BD, m∠CAB > m∠BAD, and we have the inequality: 31 > 2x - 3                   x < 17 34 > 2x                   17 > x Now, we should also guarantee that the angle is not negative or zero. 2x - 3 > 0 x > 3/2 or 1.5 Final Answer: 1.5 < x < 17

3. Solution: ΔABE and ΔDEC satisfy the Converse Hinge Theorem conditions. Since AE > CD, we know m∠ABE > m∠CED. 112 > 5x + 7 105 > 5x 21 > x Making sure the angle is not negative or zero we have: 5x + 7 > 0 x > -7/5 or -1.4 Final Answer: -1.4 < x < 21 Write an inequality, or set of inequalities, to describe the possible values for x.

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.

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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.

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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.

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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.

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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.

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