(A) Unit 3: Congruence & Constructions (SSS, SAS, ASA, AAS, HL)

Geometric Constructions

"Construction" in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge (ruler) and a pencil.

Congruent Triangles

2 triangles are congruent if and only if all pairs of corresponding sides and angles are congruent

Angle Bisector

A line, line segment, or ray that passes through the midpoint of a line segment

Congruent Angles

A line, line segment, or ray that passes through the midpoint of a line segment

Segment Bisector

A line, line segment, or ray that passes through the midpoint of a line segment

Isometry

A transformation that does not change the size or shape of a figure. A transformation in which the pre-image and image are congruent

Included Angle

An angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides AB and BC for the top triangle in the image.

CPCTC

CPCTC is an abbreviation for the phrase "Corresponding Parts of Congruent Triangles are Congruent." It can be used as a justification in a proof after you have proven two triangles congruent. BEWARE: You must have congruent triangles before you can use CPCTC.

Congruent Polygons

Congruent if and only if their corresponding angles and sides are congruent. Example: Triangles that are the same size and shape are congruent.

Corresponding Parts

Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

6. SSA

NOT a congruency postulate

7. AAA

NOT a congruency postulate

Congruent Segments

Segments that have equal lengths

5. HL: Hypotenuse-Leg Congruence Postulate

States that two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle. This postulate uses corresponding parts to prove triangles congruent.

1. SSS: Side-Side-Side Congruence Postulate

States that two triangles are congruent if all three sides of one triangle are congruent to all three sides of another triangle. This postulate uses corresponding parts to prove triangles congruent.

4. AAS: Angle-Angle-Side Congruence Postulate

States that two triangles are congruent if two angles and one of the non-included sides of one triangle are congruent to two angles and one of the non-included sides of another triangle. This postulate uses corresponding parts to prove triangles congruent.

3. ASA: Angle-Side-Angle Congruence Postulate

States that two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. This postulate uses corresponding parts to prove triangles congruent.

2. SAS: Side-Angle-Side Congruence Postulate

States that two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. This postulate uses corresponding parts to prove triangles congruent.

Vertical Angles

The angles opposite each other when two lines cross. They are always equal. Has NOTHING to do with horizontal or vertical direction.

Included Side

The common side of two consecutive angles in a polygon.

Reflexive Property

The reflexive property of equality simply states that a value is equal to itself. a = a

Rigid Motion

Transformations that do not change the size or shape of the figure; include translations, rotations, and reflections. Two polygons that are congruent can always be mapped onto one another through a series of rigid transformations.

Adjacent Triangles

Triangles that share a side, so you can apply the Reflexive Property to get a pair of congruent parts.

Congruency Statement

ΔABC ≅ ΔDEF The order of the letters is very important, as corresponding parts are written in corresponding order. In the image, angle D is congruent to angle G because they each are the first letters of how they are named. Notice that the congruent sides also line up within the congruence statement.