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Two Sides And The Angle Between

Equal angles only fix the shape. Two sides and the angle between them fix the whole triangle: the SAS rule.

11 micro-lessons

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What If They Measure the Angles?

Meera and Rabia try measuring three angles instead of three sides.

Suppose Meera and Rabia measure the three angles of the frame instead of the three sides, getting $30^\circ$ , $70^\circ$ , and $80^\circ$ . Can they make an exact copy of the triangle with just this information?

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Same Angles, Three Sizes

Three triangles with angles 30, 70 and 80 degrees at different sizes.

$Two Sides and the Angle Between$

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same angles, different sizes.

Angles Fix Shape, Not Size

Equal angles give similar-looking triangles that can be any size.

As we just saw, triangles with identical angle measurements have the same shape, but not the same size. Hence, two triangles that have the same set of angles need not be congruent.

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Two Sides Plus the Angle Between

The SAS case: two sides and the angle enclosed by them.

Consider $\Delta ABC$ and $\Delta XYZ$ where $AB = XY = 6\text{ cm}$ , $AC = XZ = 5\text{ cm}$ , and $\angle A = \angle X = 30^\circ$ . Do these measurements guarantee congruence?

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Building the SAS Triangle

The one-and-only triangle from 6 cm, 30 degrees and 5 cm.

$Two Sides and the Angle Between$

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Constructing two sides and the included angle.

SAS Condition Card

The SAS test, with the included-angle requirement spelled out.

SAS (Side Angle Side) Condition

What You Need

Two pairs of equal sides and the pair of equal angles LYING BETWEEN them.

$A B = X Y$ , $A C = XZ$ , $∠ A = ∠ X$

Conclusion

$Δ A BC ≅ Δ X Y Z$

Units

cm for sides, degrees for angles.

Condition and Scope

The angle must be the included one. If the equal angle is not between the two equal sides, this becomes the SSA case, and the congruence test does not apply.

Example Substitution

A triangle with sides 6 cm and 5 cm, and a 30° angle between them.

The Crossing Segments at O

Using SAS where O is the midpoint of both AD and BC.

Problem. In a given figure, segments $AD$ and $BC$ cross at $O$ , with $A$ top-left, $C$ top-right, $B$ bottom-left, and $D$ bottom-right. It is given that $O$ is the midpoint of $AD$ and $O$ is the midpoint of $BC$ . What can be said about the lengths $AB$ and $CD$ ?

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← previousNaming Congruent Triangles next →Why SSA Is Not Enough

What If They Measure the Angles?

Meera and Rabia try measuring three angles instead of three sides.

1 / 2

Two Sides Plus the Angle Between

The SAS case: two sides and the angle enclosed by them.

Consider $\Delta ABC$ and $\Delta XYZ$ where $AB = XY = 6\text{ cm}$ , $AC = XZ = 5\text{ cm}$ , and $\angle A = \angle X = 30^\circ$ . Do these measurements guarantee congruence?

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The Crossing Segments at O

Using SAS where O is the midpoint of both AD and BC.

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