Triangle Similarity and Congruence Theorems

Questions and Answers

What does the Angle-Angle Similarity Postulate (AA) state?

Two triangles are similar if they have two corresponding angles that are equal. (correct)

Two triangles must have at least one angle congruent.

Two triangles are similar if their corresponding sides are parallel.

Two triangles are always similar.

What does the Side-Side-Side Similarity Theorem (SSS) state?

If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar.

What is the meaning of SAS in triangle similarity?

Side, Angle, Side.

The Side-Angle-Side Congruence Postulate (SAS) states that two triangles are similar if two sides and the included angle are congruent.

False Signup and view all the answers

What is the condition for the Side-Side-Side Congruence Postulate (SSS)?

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Signup and view all the answers

What does the Angle-Side-Angle Congruence Postulate (ASA) require?

Two angles and the included side of one triangle must be congruent to two angles and the included side of another triangle. Signup and view all the answers

What does the Triangle Proportionality Theorem state?

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Signup and view all the answers

The Converse of the Triangle Proportionality Theorem states that if a line divides two sides of a triangle proportionally, it is parallel to the third side.

True Signup and view all the answers

What is the focus of the Triangle Angle Bisector Theorem?

It concerns the relative lengths of the segments created by a triangle's angle bisector. Signup and view all the answers

What does the Hypotenuse-Leg Congruence Theorem (HL) state?

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Signup and view all the answers

If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle.

True Signup and view all the answers

Study Notes

Triangle Similarity Theorems

Angle-Angle Similarity Postulate (AA): Two triangles are similar if two corresponding angles are congruent.
Side-Side-Side Similarity Theorem (SSS): Triangles are similar if the lengths of their corresponding sides are proportional.

Triangle Congruence Theorems

Side-Angle-Side Congruence Postulate (SAS): If two sides and the included angle of one triangle are congruent to those of another triangle, they are congruent.
Side-Side-Side Congruence Postulate (SSS): Triangles are congruent if all three sides of one triangle are congruent to the corresponding sides of another triangle.
Angle-Side-Angle Congruence Postulate (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, they are congruent.
Angle-Angle-Side Congruence Postulate (AAS): If two angles and the non-included side of one triangle are congruent to those of another triangle, the triangles are congruent.

Proportional Relationships in Triangles

Triangle Proportionality Theorem: A line parallel to one side of a triangle that intersects the other two sides divides those sides proportionally.
Converse of the Triangle Proportionality Theorem: A line that divides two sides of a triangle proportionally is parallel to the third side.

Triangle Angle Bisector Theorem: The lengths of the segments created by an angle bisector on the opposite side are proportional to the lengths of the other two sides of the triangle.
Hypotenuse-Leg Congruence Theorem (HL): In right triangles, if the hypotenuse and one leg are congruent to those of another right triangle, the triangles are congruent.

Additional Theories

No Name Similarity Theorem: When an altitude is drawn to the hypotenuse of a right triangle, the two triangles formed are similar to the original triangle and to each other.

Description

Test your knowledge on the various similarity and congruence theorems related to triangles. This quiz covers key concepts such as the Angle-Angle Similarity Postulate and the Side-Angle-Side Congruence Postulate. Perfect for students looking to solidify their understanding of triangle properties.