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Definition of a parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.

Properties of a parallelogram:

1. opposite sides are congruent

2. opposite angles are congruent

3. consecutive angles are supplementary (180 degrees)

4. diagonals bisect each other.

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Rectangles have the same properties of parallelograms:

1. Opposite sides are congruent.

2. Opposite sides are parallel.

3. Opposite angles are congruent.

4. Consecutive angles are supplementary (180 degrees)

5. Diagonals bisect each other.

Plus the following:

6. All 4 angles are right angles.

7. Diagonals are congruent.

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Rhombi have the same properties of parallelograms:

1. Opposite sides are congruent.

2. Opposite sides are parallel.

3. Opposite angles are congruent.

4. Consecutive angles are supplementary.

5. Diagonals bisect each other.

Plus these

6. All sides are congruent.

7. Diagonals are perpendicular.

8. Diagonals bisect angles on half

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Properties of squares:

A square has ALL the properties of a parallelogram, rectangle, and rhombus.

1. Opposite sides are congruent.

2. Opposite sides are parallel.

3. Opposite angles are congruent.

4. Consecutive angles are supplementary (180 degrees).

5. Diagonals bisect each other.

6. Four right angles.

7. Diagonals are congruent.

8. Four congruent sides.

9. Diagonals are perpendicular.

10. Diagonals bisect opposite angles.

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Non-isosceles trapezoid properties:

1. Only ONE pair of opposite sides are parallel.

2. Consecutive angles between parallel lines are supplementary (180 degrees).

Isosceles trapezoid properties:

Isosceles trapezoids have the same properties as non-isosceles trapezoids, plus these:

1. Non-parallel sides (legs) are congruent.

2. Diagonals are congruent.

3. Base angles are congruent.

4. Opposite angles are supplementary.

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A kite is a quadrilateral with the following properties:

1. Exactly 2 pairs of consecutive congruent sides (segment AB is equal to segment AD and segment BC is equal to segment DC).

2. One pair of opposite angles are congruent. (angle ABC is equal to angle ADC).

3. Diagonals are perpendicular (segment AC is perpendicular to segment BD).

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Parallel definition: lines that never intersect.

Perpendicular definition: opposite reciprocals.

Examples would be: 4/5, -5/4, 2/6, -6,2 (X2-X1), (Y2-Y1)

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Point: It is a location which has no size or shape.

Line: It is made up of infinite points, where at least 2 form a line. A line has no width or height.

Plane: A plane is a flat surface consisting/ made up of points. Any 3 points create a plane. A plane goes in all directions forever.

Distance formula:

Formula: D= d=(x2​−x1​)2+(y2​−y1​)2​

Midpoint formula:

M=(2x1​+x2​​,2y1​+y2​​)

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Venn diagrams: There are two circles, and their arrangement shows the relationship between two variables. For example, if circle p is entirely inside circle q, it means that all the elements of p are also elements of q.

Conditional statements: A conditional statement links two ideas, stating that if the first idea is true, then the second idea must also be true, and it may involve "and" (both conditions must be true) or "or" (at least one condition must be true).

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Point slope formula: y-y1=m(x-x1)

When given a graph it is necessary to be able to identify the slope. M=rise/run meaning y over x. When provided ordered pairs you use, m= (y1-y2) / (x1-x2).

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The midsegment of a trapezoid is the line that connects the midpoints of its two non-parallel sides. It is always parallel to the two bases and its length is the average of the lengths of the bases. This means you add the lengths of the bases and divide by 2 to find the midsegment's length.

For example:

A trapezoid has bases measuring 8 units and 12 units.

To find the midsegment: Midsegment=Base 1+Base 22=8+122=10 units.

Midsegment=2Base 1+Base 2​=28+12​=10 units.

The midsegment is 10 units long.

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Triangle congruence means two triangles are identical in both size and shape, with corresponding sides and angles being equal. You can prove they are congruent using methods like SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), or HL (hypotenuse-leg) for right triangles.

Example:

Given:

Triangle ABC and triangle DEF.

AB = DE = 5 cm, BC = EF = 7 cm, and ∠B = ∠E = 60°.

Steps to Prove Congruence Using SAS:

1. Side 1: AB = DE = 5 cm (Given)

2. Angle: ∠B = ∠E = 60° (Given)

3. Side 2: BC = EF = 7 cm (Given)

Because 2 sides and the included angle between them are the same in both triangles, by the SAS (Side-Angle-Side) criterion, the triangles are congruent.

Thus, △ABC ≅ △DEF.

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Triangle Congruence Theorems, Two Column Proofs, SSS, SAS, ASA, AAS Postulates, Geometry  Problems