# Write a congruence statement for the congruent triangles in each diagram

This establishes that it is reasonable to take the SSS congruence test as an axiom of geometry.

## Congruence statements and corresponding parts

This construction has yielded two triangles with the given measurements. Notice that this congruence test tells us that the three angles of a triangle are completely determined by its three sides. These notes will introduce and discuss them one by one, but they, and their standard initials, are summarised here at the start to indicate the structure of the following discussion: The four standard congruence tests for triangles Two triangles are congruent if: SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or SAS: two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle, or AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or RHS: the hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one side of the other right-angled triangle. There are a few possible cases: If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side SSA, or long side-short side-angle , then the two triangles are congruent. However, in spherical geometry and hyperbolic geometry where the sum of the angles of a triangle varies with size AAA is sufficient for congruence on a given curvature of surface. ASA Angle-Side-Angle : If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. However, these two triangles, are congruent. Constructing a triangle with three given sides When all three sides of a triangle are given, however, there is no longer any freedom of movement, and only one such triangle can be constructed up to congruence.

The opposite side is sometimes longer when the corresponding angles are acute, but it is always longer when the corresponding angles are right or obtuse. This shows that just knowing that two pairs of sides are equal is not enough information to establish congruence.

Determining congruence Sufficient evidence for congruence between two triangles in Euclidean space can be shown through the following comparisons: SAS Side-Angle-Side : If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

The example below shows that a quadrilateral with opposite sides equal is a parallelogram.

There are a few possible cases: If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side SSA, or long side-short side-anglethen the two triangles are congruent.

The shape of a triangle is determined up to congruence by specifying two sides and the angle between them SAStwo angles and the side between them ASA or two angles and a corresponding adjacent side AAS. This is the ambiguous case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence.

### Congruence statement example

Determining congruence Sufficient evidence for congruence between two triangles in Euclidean space can be shown through the following comparisons: SAS Side-Angle-Side : If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. However, in spherical geometry and hyperbolic geometry where the sum of the angles of a triangle varies with size AAA is sufficient for congruence on a given curvature of surface. The example below shows that a quadrilateral with opposite sides equal is a parallelogram. Specifying two sides and an adjacent angle SSA , however, can yield two distinct possible triangles. If we are given the lengths of the three sides of a triangle, then only one such triangle can be constructed up to congruence. If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent. These notes will introduce and discuss them one by one, but they, and their standard initials, are summarised here at the start to indicate the structure of the following discussion: The four standard congruence tests for triangles Two triangles are congruent if: SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or SAS: two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle, or AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or RHS: the hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one side of the other right-angled triangle. If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle but less than the length of the adjacent side , then the two triangles cannot be shown to be congruent. The shape of a triangle is determined up to congruence by specifying two sides and the angle between them SAS , two angles and the side between them ASA or two angles and a corresponding adjacent side AAS. ASA Angle-Side-Angle : If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Notice that this congruence test tells us that the three angles of a triangle are completely determined by its three sides. What restriction must be placed on the three side lengths in order for a triangle with those side lengths to exist? This construction has yielded two triangles with the given measurements. This establishes that it is reasonable to take the SSS congruence test as an axiom of geometry. This is the ambiguous case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence.

ASA Angle-Side-Angle : If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

Three such triangles are shown below, and they are clearly not congruent.

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