Lines, Angles, and Polygons

Vertical Angles: When two lines intersect, the angles opposite each other are called vertical angles. Vertical angles are always equal in measure.

Complementary Angles: Two angles are complementary if the sum of their measures is 90 degrees. For example, if one angle measures 30 degrees, its complement measures 60 degrees.

Supplementary Angles: Two angles are supplementary if the sum of their measures is 180 degrees. For example, if one angle measures 120 degrees, its supplement measures 60 degrees.

Linear Pair: A linear pair consists of two adjacent angles whose non-common sides form a straight line. The angles in a linear pair are always supplementary.

An angle bisector is a line or ray that divides an angle into two equal parts. If a line bisects an angle, it creates two angles of equal measure.

Example: If a line bisects an angle of 80 degrees, it creates two angles of 40 degrees each.

A line bisector is a line that divides another line segment into two equal parts. The point where the bisector intersects the line segment is called the midpoint.

Example: If a line bisects a line segment of length 10 units, it creates two segments of 5 units each.

When a transversal (a line that intersects two or more other lines) crosses parallel lines, several special angle relationships are formed:

• Corresponding angles are equal in measure.
• Alternate interior angles are equal in measure.
• Alternate exterior angles are equal in measure.
• Same-side interior angles are supplementary (their sum is 180 degrees).
• Same-side exterior angles are supplementary (their sum is 180 degrees).

While you don't need to memorize the terminology, understanding these relationships is crucial for solving problems involving parallel lines and transversals.

Example: If two parallel lines are cut by a transversal, and one of the angles formed measures 70 degrees, you can determine the measures of the other angles using these relationships.

Triangle: The sum of the interior angles of a triangle is always 180 degrees.

Quadrilateral: The sum of the interior angles of a quadrilateral is always 360 degrees.

General Formula: For a polygon with $n$ sides, the sum of the interior angles is:

$180(n-2)$

Regular Polygons: In a regular polygon (where all sides and angles are equal), the measure of each interior angle is:

$\frac{180}{n}(n-2)$

Example: In a regular hexagon (6 sides), the sum of the interior angles is $180(6-2)=720$ degrees, and each interior angle measures $\frac{720}{6}=120$ degrees.

An exterior angle of a polygon is formed by extending one side of the polygon and measuring the angle between the extended side and the adjacent side.

Sum of Exterior Angles: The sum of the exterior angles of any polygon is always 360 degrees, regardless of the number of sides.

Regular Polygons: In a regular polygon, each exterior angle measures:

$\frac{360}{n}$

Example: In a regular pentagon (5 sides), each exterior angle measures $\frac{360}{5}=72$ degrees.

Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional.

Properties of Similar Polygons:

• Corresponding angles are equal.
• Corresponding sides are proportional.
• The ratio of their perimeters is equal to the ratio of their corresponding sides.
• The ratio of their areas is equal to the square of the ratio of their corresponding sides.

Example: If two triangles are similar with a ratio of corresponding sides of 2:3, then the ratio of their areas is 4:9.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Parallelograms have several important properties:

• Opposite sides are equal in length.
• Opposite angles are equal in measure.
• Consecutive angles are supplementary (their sum is 180 degrees).
• The diagonals bisect each other.

Rectangle: A parallelogram with four right angles. In addition to the properties of a parallelogram, a rectangle has:

• All angles are right angles (90 degrees).
• The diagonals are equal in length.

Rhombus: A parallelogram with all sides equal in length. In addition to the properties of a parallelogram, a rhombus has:

• All sides are equal in length.
• The diagonals are perpendicular to each other.
• The diagonals bisect the angles.

Square: A parallelogram that is both a rectangle and a rhombus. A square has all the properties of both a rectangle and a rhombus:

• All sides are equal in length.
• All angles are right angles (90 degrees).
• The diagonals are equal in length and perpendicular to each other.
• The diagonals bisect the angles.

Trapezoid: A quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases, and the non-parallel sides are called the legs.

Create or Redraw Diagrams: Whenever possible, create or redraw diagrams on your scratch paper. This helps you visualize the problem and identify relationships between different elements.

Extend Lines: It often helps to draw imaginary lines extending from polygons. This can reveal hidden relationships and make it easier to apply geometric principles.

Create Triangles: When you can, try to create triangles within more complex shapes. Triangles have well-defined properties that can be used to solve problems.

When a problem involves multiple triangles, look for them to be similar. Similar triangles have proportional sides and equal angles, which can be used to find unknown measurements.

Example: If you have two triangles with equal angles, they are similar, and you can set up a proportion to find unknown sides.

Remember the key angle relationships:

• Vertical angles are equal.
• Complementary angles sum to 90 degrees.
• Supplementary angles sum to 180 degrees.
• Angles in a triangle sum to 180 degrees.
• Angles in a quadrilateral sum to 360 degrees.

These relationships can often be used to find unknown angle measures.

Complex geometry problems can often be broken down into simpler parts. Look for:

• Right triangles, which allow you to use the Pythagorean theorem.
• Special triangles (30-60-90, 45-45-90), which have known side ratios.
• Similar figures, which have proportional sides.
• Congruent figures, which have equal sides and angles.

By identifying these simpler parts, you can often solve complex problems step by step.