Glossary
AA (Angle-Angle) Similarity
A criterion stating that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
Example:
If you can show that two angles in one triangle match two angles in another, you've proven AA Similarity.
AAS (Angle-Angle-Side) Congruence
A criterion stating that if two angles and a non-included side of one triangle are equal to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
Example:
If you know two angles and a side not between them are equal in two triangles, you can apply AAS Congruence to confirm they are exact copies.
ASA (Angle-Side-Angle) Congruence
A criterion stating that if two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.
Example:
When building a bridge, engineers might use ASA Congruence to ensure two triangular supports are identical by checking two angles and the side connecting them.
Alternate Exterior Angles
Angles that are on opposite sides of the transversal and outside the two parallel lines; these angles are congruent.
Example:
When a transversal cuts parallel lines, the angles in the top-right and bottom-left positions outside the parallel lines are Alternate Exterior Angles.
Alternate Interior Angles
Angles that are on opposite sides of the transversal and between the two parallel lines; these angles are congruent.
Example:
If you draw a 'Z' shape across two parallel lines, the angles inside the 'Z' at the corners are Alternate Interior Angles.
Area Relationships (Similarity)
For similar figures with a scale factor 'k', the ratio of their areas is k².
Example:
If a small square has an area of 4 and a larger similar square has a Scale Factor of 3, its area will be 4 * 3² = 36.
Congruent Figures
Figures that are identical in both size and shape, meaning they are perfect copies of each other.
Example:
If you trace a triangle on paper and then cut it out, the traced triangle and the cutout triangle are Congruent Figures.
Corresponding Angles
Angles in congruent or similar figures that are in the same relative position and have equal measures.
Example:
When two triangles are similar, the angle at the top vertex of one triangle is the Corresponding Angle to the angle at the top vertex of the other.
Corresponding Angles (Parallel Lines)
Angles that are in the same relative position at each intersection when a transversal line crosses two parallel lines; these angles are congruent.
Example:
Imagine two parallel train tracks crossed by a road; the angles in the top-left position at each intersection are Corresponding Angles.
Corresponding Sides
Sides in congruent or similar figures that are in the same relative position and are either equal in length (congruent figures) or proportional (similar figures).
Example:
In two similar rectangles, the longest side of one rectangle is the Corresponding Side to the longest side of the other.
Exterior Angle Theorem
A theorem stating that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles.
Example:
If a triangle has interior angles of 40° and 60°, the Exterior Angle Theorem tells us the exterior angle at the third vertex is 40° + 60° = 100°.
Hypotenuse
The longest side of a right triangle, always located directly opposite the right angle.
Example:
When climbing a ladder leaning against a wall, the ladder itself forms the Hypotenuse of the right triangle created with the wall and the ground.
Legs
The two shorter sides of a right triangle that form the right angle.
Example:
In a right triangle, the two sides that meet at the 90-degree corner are called the Legs.
Pythagorean Theorem
A fundamental theorem in geometry stating that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b), expressed as a² + b² = c².
Example:
If a right triangle has Legs of length 3 and 4, you can use the Pythagorean Theorem to find the hypotenuse: 3² + 4² = c², so 9 + 16 = 25, and c = 5.
Rigid Transformations
Geometric transformations (translations, rotations, and reflections) that move a figure without changing its size or shape, resulting in a congruent image.
Example:
Sliding a book across a table is an example of a Rigid Transformation (a translation), as the book's size and shape remain unchanged.
SAS (Side-Angle-Side) Congruence
A criterion stating that if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
Example:
If you have two triangles where two sides and the angle between them are identical, you can use SAS Congruence to prove they are the same.
SAS (Side-Angle-Side) Similarity
A criterion stating that if the ratios of the lengths of two corresponding sides are equal and the included angles are equal, then the triangles are similar.
Example:
If two triangles have proportional sides and the angle between those sides is the same, you can use SAS Similarity to show they are scaled versions of each other.
SSS (Side-Side-Side) Congruence
A criterion stating that if all three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the triangles are congruent.
Example:
To prove two triangles are congruent using SSS Congruence, you'd measure all three sides of each triangle and confirm they match up perfectly.
SSS (Side-Side-Side) Similarity
A criterion stating that if the ratios of the lengths of corresponding sides of two triangles are equal, then the triangles are similar.
Example:
To prove SSS Similarity, you would divide the length of each side of the first triangle by its corresponding side in the second triangle and confirm all ratios are identical.
Same-Side Exterior Angles
Angles that are on the same side of the transversal and outside the two parallel lines; these angles are supplementary (add up to 180°).
Example:
Consider two parallel streets and a crosswalk; the angles on the same side of the crosswalk but outside the streets are Same-Side Exterior Angles.
Same-Side Interior Angles
Angles that are on the same side of the transversal and between the two parallel lines; these angles are supplementary (add up to 180°).
Example:
If you draw a 'U' shape across two parallel lines, the angles inside the 'U' at the corners are Same-Side Interior Angles.
Scale Factor (k)
The ratio of the lengths of corresponding sides in similar figures, indicating how much one figure has been scaled up or down relative to another.
Example:
If a map has a Scale Factor of 1:100,000, then 1 unit on the map represents 100,000 units in real life.
Similar Figures
Figures that have the same shape but can be different sizes, meaning they are scaled versions of each other.
Example:
A small photograph and a larger print of the same image are Similar Figures because they have the same proportions but different dimensions.
Triangle Sum Theorem
A fundamental theorem stating that the sum of the interior angles in any triangle is always 180 degrees.
Example:
If you know two angles of a triangle are 60° and 70°, you can use the Triangle Sum Theorem to find the third angle: 180° - 60° - 70° = 50°.
Vertical Angles
A pair of angles formed by two intersecting lines that are opposite each other and are always congruent (equal in measure).
Example:
When two roads cross, the angles directly across from each other at the intersection are Vertical Angles and will always be equal.
Volume Relationships (Similarity)
For similar 3D figures with a scale factor 'k', the ratio of their volumes is k³.
Example:
If a small cube has a volume of 8 and a larger similar cube has a Scale Factor of 2, its volume will be 8 * 2³ = 64.