KA Geometry Unit 3: Congruence

How can we determine if two shapes are congruent?

By seeing if one shape can be mapped onto the other using rigid transformations.

Which transformation is NOT a rigid transformation?

Dilation.

What are rigid transformations?

Transformations that preserve distance between points.

What is congruence?

Shapes with same shape and size

If AB = CD, are AB and CD congruent?

Yes, because AB can be mapped onto CD with a rigid transformation.

When are angles congruent?

ONLY if they have the same measure.

If ABC ≅ DEF, what does this mean about their angle measures?

Rigid transformations preserve angle measures, so corresponding angles are congruent.

What does the SSS Triangle Congruence Criterion state?

If three side lengths of two triangles are corresponding and congruent, then the triangles are congruent.

What is a Triangle Congruence Criterion?

A way of proving that two triangles are congruent.

What are the four main Triangle Congruence Criteria?

SSS, SAS, ASA, AAS.

What does SAS stand for, and what does it mean?

Side, Angle, Side. Two sides and the included angle (the angle between them) are congruent.

What is the difference between SAS and SSA?

In SAS, the angle is between the two sides. In SSA, the angle is not between the two sides.

Does SSA prove congruence?

No.

What does ASA stand for, and what does it mean?

Angle, Side, Angle. Two angles and the included side (the side between them) are congruent.

What does AAS stand for, and what does it mean?

Angle, Angle, Side. Two angles and a non-included side are congruent.

Does AAA prove congruence?

No.

What does AAA imply?

Similarity.

What is similarity?

Same shape, but not necessarily the same size.

Are all congruent figures similar?

Yes.

Are all similar figures congruent?

No.

What are the four Triangle Congruence Theorems that work?

SSS, SAS, ASA, AAS.

What is the Hypotenuse-Leg (HL) theorem?

If the hypotenuse and a leg of two right triangles are congruent, then the triangles are congruent.

What Triangle Congruence Theorems DO NOT work?

AAA and SSA.

How do you map corresponding vertices in congruent triangles?

Corresponding vertices are listed in the order of the congruence statement (e.g., ABC ≅ NMO means A corresponds to N, B to M, and C to O).

If all angles are congruent but no side lengths are known, can you prove congruence?

No, you need at least one side length.

What is congruence the equivalence of for shapes?

Same size and same shape.

What do rigid transformations preserve?

Congruence.

What are corresponding sides?

Sides that correspond and are congruent to each other.

What is a proof?

A formal description of how we know something is true based on mathematical information.

What are common reasons used in two-column proofs?

Geometric theorems, algebraic properties, and given information.

When a transversal crosses parallel lines, what angles are congruent?

Alternate interior angles.

If four distinct angles are given, can you always prove triangle congruence?

No.

If all angles are congruent but no side lengths are given, can you prove congruence?

No.

What is the Reflexive Property?

A quantity is equal (or congruent, or similar) to itself.

When is the Reflexive Property commonly used?

When shapes share sides or angles.

State the Addition Property of Equality.

If a = b, then a + c = b + c.

State the Subtraction Property of Equality.

If a = b, then a - c = b - c.

State the Multiplication Property of Equality.

If a = b, then a * c = b * c.

State the Division Property of Equality.

If a = b, then a / c = b / c (where c ≠ 0).

State the Substitution Property.

If a = b, then a may be replaced by b in any equation or expression.

State the Distributive Property.

a(b + c) = ab + ac.

State the Transitive Property.

If a = b and b = c, then a = c.

State the Symmetric Property.

If a = b, then b = a.

What is the difference between equality and congruence?

Equality is for numerical values; congruence is for geometric figures.

What is an isosceles triangle?

A triangle with two congruent sides.

What is the relationship between the base angles of an isosceles triangle?

They are congruent.

If two sides of a triangle are congruent, what can you conclude about the angles opposite those sides?

They are congruent.

What is an equilateral triangle?

A triangle with all three sides congruent.

What is the measure of each angle in an equilateral triangle?

60 degrees.

What is the sum of the angles in a triangle?

180 degrees.

What are supplementary angles?

Two angles whose measures add up to 180 degrees.

What is the Exterior Angle Theorem?

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

In a right triangle, if you know the lengths of two sides, how can you find the third side?

Use the Pythagorean theorem.

In congruent triangles, what is the relationship between corresponding angles?

They are congruent.

How can you find the measure of angles in isosceles and equilateral triangles?

Use the properties of these triangles (base angles are congruent in isosceles, all angles are 60 degrees in equilateral) and the fact that the angles in a triangle sum to 180 degrees.

If you are given the exterior angle of a triangle, how can you find the remote interior angles?

The exterior angle is equal to the sum of the two remote interior angles.

What are the properties of a parallelogram?

Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other.

What are the properties of a rectangle?

All properties of a parallelogram, all angles are right angles, diagonals are congruent.

What are the properties of a square?

All properties of a parallelogram, rectangle, and rhombus; all sides are congruent, all angles are right angles, diagonals are congruent and perpendicular, diagonals bisect opposite angles.

What are the properties of a trapezoid?

One pair of opposite sides is parallel (bases).

What are the properties of an isosceles trapezoid?

All properties of a trapezoid, non-parallel sides (legs) are congruent, base angles are congruent, diagonals are congruent.

What are the properties of a kite?

Two pairs of adjacent sides are congruent, diagonals are perpendicular, one diagonal bisects the other, one pair of opposite angles is congruent.

What do you know about the sides of a triangle if a line segment connects the midpoints of two sides?

The line segment is parallel to the third side and half its length.

What are vertical angles?

Two angles formed by intersecting lines that are opposite each other.

What is the relationship between vertical angles?

They are congruent.

If two line segments are congruent, what can you deduce about other angles or sides?

Depending on the context, you might be able to use congruence criteria (SSS, SAS, ASA, AAS) or other geometric theorems to deduce further relationships. More information is needed.

What is the definition of constructing congruent angles?

Creating an angle with the same measure as a given angle using only a compass and straightedge.

What are the steps to construct congruent angles?

(See your notes for the detailed steps.)

What is the definition of constructing a parallel line?

Drawing a line through a given point that does not intersect the original line.

What are the steps to construct a parallel line?

(See your notes for the detailed steps.)

What is the definition of constructing a perpendicular line through a point on the line?

Creating a perpendicular line passing through a given point that lies on the original line.

What are the steps to construct a perpendicular line through a point on the line?

(See your notes for the detailed steps.)

What is the definition of constructing a perpendicular line through a point not on the line?

Creating a perpendicular line passing through a point that is not on the original line.

What are the steps to construct a perpendicular line through a point not on the line?

(See your notes for the detailed steps.)

What is the definition of constructing an angle bisector?

Dividing an angle into two equal parts.

What are the steps to construct an angle bisector?

(See your notes for the detailed steps.)

What are the properties of a rhombus?

All properties of a parallelogram, all sides are congruent, diagonals are perpendicular, diagonals bisect opposite angles.