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Two Angles And One Side

Two angles and the side between them (ASA), and how the 180° angle sum turns other cases into ASA (AAS).

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Now Try Two Angles and a Side

Sides and one angle have been tested; what about angles and one side?

We have explored several conditions for checking if two triangles are identical. We know that checking all three sides (SSS) works, and checking two sides and the angle between them (SAS) works. However, two sides and a non-included angle (SSA) does not always guarantee congruence.

Now, what if we reverse the SAS scenario? If we have examined cases using two sides and an angle, can we use two angles and a side?

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The ASA Condition

Two angles with the side between them fix the triangle completely.

To check if non-congruent triangles can exist with two angles and an included side, let's walk through the construction of the triangle:

First, draw the base line segment $BC = 5\text{ cm}$ .
At point $B$ , draw a ray making an angle of $50^\circ$ with $BC$ .
At point $C$ , draw a ray making an angle of $30^\circ$ with $CB$ on the same side.

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Two Rays, One Meeting Point

The ASA construction on a 5 cm base with 50 and 30 degree angles.

$Two Angles and One Side$

Click to zoom

The two angled rays can only intersect in exactly one place, guaranteeing a unique triangle.

ASA Condition Card

The ASA test and how to check the side is the included one.

ASA (Angle Side Angle) Condition

What You Need:

Two pairs of equal angles: $∠ B = ∠ Y$ and $∠ C = ∠ Z$
The pair of equal sides lying BETWEEN them: $BC = Y Z$

Conclusion:

Δ A BC ≅ Δ X Y Z

Conditions & Scope

The side must join the vertices of the two named angles. If it does not, use the 180° angle sum to find the third angle first.

Example Substitution

BC = 5 cm

with

5 0^{\circ}

at B and

3 0^{\circ}

at C.

Corresponding Vertices at O

Reading the correct vertex matching from the crossing-segments figure.

Problem

In a figure where diagonals cross at point $O$ , $O$ is the midpoint of both $AD$ and $BC$ . We know that $AO = OD$ and $BO = OC$ . The vertically opposite angles $\angle AOB$ and $\angle DOC$ are equal. Using the SAS condition, we confirmed the triangles are congruent. Name the corresponding vertices of the two triangles and express the congruence to find the relationship between $AB$ and $DC$ .

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Filling In the Third Angle

Fig. 1.2 before and after the missing 70 degree angles are found.

$Two Angles and One Side$

Click to zoom

When a side is not between the given angles, finding the third angle allows us to use the ASA condition.

When the Side Is Not Between

Turning 35 degrees, 75 degrees and BC = 4 cm into an ASA situation.

Problem

$\Delta ABC$ and $\Delta XYZ$ are such that $\angle A = \angle X = 35^\circ$ , $\angle C = \angle Z = 75^\circ$ , and $BC = YZ = 4\text{ cm}$ . Are they congruent?

Notice that the side $BC$ lies between angles $B$ and $C$ , but the angles given are $A$ and $C$ . Because the side is NOT included between the given angles, ASA does not apply directly.

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← previousWhy SSA Is Not Enough next →Right Triangles And RHS

The ASA Condition

Two angles with the side between them fix the triangle completely.

To check if non-congruent triangles can exist with two angles and an included side, let's walk through the construction of the triangle:

First, draw the base line segment $BC = 5\text{ cm}$ .
At point $B$ , draw a ray making an angle of $50^\circ$ with $BC$ .
At point $C$ , draw a ray making an angle of $30^\circ$ with $CB$ on the same side.

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Corresponding Vertices at O

Reading the correct vertex matching from the crossing-segments figure.

Problem

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When the Side Is Not Between

Turning 35 degrees, 75 degrees and BC = 4 cm into an ASA situation.

Problem

$\Delta ABC$ and $\Delta XYZ$ are such that $\angle A = \angle X = 35^\circ$ , $\angle C = \angle Z = 75^\circ$ , and $BC = YZ = 4\text{ cm}$ . Are they congruent?

Notice that the side $BC$ lies between angles $B$ and $C$ , but the angles given are $A$ and $C$ . Because the side is NOT included between the given angles, ASA does not apply directly.

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