Intro to geome🌳 + proofs

i am still patting myself on the back for thinking of the emoji pun.

Point

An exact location or position on a plane. Has no size.

• P

Symbol for a point (point P)

Line

Collection of points extending infinitely in opposite directions. No thickness, only 1 dimension.

↔

Symbol for a line

Collinear points

Points on the same line

Segment

Part of a line with 2 defined end points and length

—

Symbol for a segment

Ray

Part of a line with 1 endpoint and extends infinitely in another direction

→

Symbol for a ray

Vertical angles

The opposite angles when two lines intersect

Vertical angles are…

Congruent

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Linear pair

2 angles that lie on the same line and share a common ray

Parallel lines

2 lines that do not and will never intersect

||

Symbol for parallel lines

Perpendicular lines

2 Lines that intersect at a 90° angle

L⊥M

Symbol for perpendicular line of L and M

Skew lines

2 Lines that do not intersect and are in different planes

Are these skew lines?

Yes

Are these skew lines?

No

Plane (face)

Set of points that form a completely flat surface, extends infinitely

Plane GCDE

How do we say the plane made out of points GCDE?

Coplanar

Points, lines, segments, rays….. on the same plane

Are skew lines coplanar?

No

Angles

Set of points inside and on two different intersecting rays

∠ABC

Symbol for angle ABC

Vertex

Point where the rays of an angle meet

Legs

Sides of an angle

Interior

Set of points inside the angle

Exterior

Set of points outside the angle

Acute angle

Angle that measures less than 90°

Right angle

An angle that measures exactly 90°

Obtuse angle

An angle that measures 90°< x <180°

Straight angle

An angle that measures exactly 180°

Reflex angle

An angle that is greater than 180°

Transversal line

A line that passes through 2 lines in the same plane at 2 distinct points

Corresponding angles

A set of angles that are in the SAME POSITION

Alternate interior angles

A set of angles on opposite sides of the transversal in the interior

Consecutive interior angles

A set of angles in the interior on the same side of the transversal

What are angles 2 and 6?

Corresponding angles

What are angles 3 and 5?

Alternate interior angles

What are angles 4 and 5?

Consecutive interior angles

If lines are parallel, corresponding angles are

Congruent

If lines are parallel, then alternate interior angles are

Congruent

If lines are parallel, then consecutive interior angles are

Supplementary

Reflexive property

A quantity is equal to itself

Example of reflexive property

a=a

Symmetric property

An equality maybe reversed

Apply the symmetric property onto a=c

c=a

Transitive property

If two different quantities are equal to the same quantity, those two quantities are equal

Transitive property example

a=b, b=c, so a=c

Substitution property

Any quantity may be substituted for its equal at any time

Adjacent angles

Angles that are next to each other, each sharing a common ray

Definition of Complementary angles (Def of comp)

Two angles that add to 90°

Which example of using Def of comp in a proof is correct?

∠1 and ∠2 are complementary angles, [some other theorem] → ∠1+∠2=90, Def of comp

Definition of supplementary angles (Def of supp)

Two angles that add to 180°

Which example of using Def of supp is correct?

∠1 and ∠2 are supplementary, [some other theorem] → ∠1+∠2=180, Def of supp

Linear pair

An adjacent set of two ∠s on a line that are supplementary

Congruent supplements theorem

When 2 ∠s are supplementary to the same ∠, they are ≅

Vertical angle theorem

Vertical angles are ≅

Congruent complements theorem

When 2 ∠s are complementary to the same ∠, they are ≅

Corresponding angle theorem (corr ∠ theorem)

If lines are parallel, then corresponding angles are congruent

Alternate interior angle theorem (alt int ∠ theorem)

If lines are parallel, then alternate interior angles are congruent

Consecutive interior angle theorem (cons int ∠ theorem)

If lines are parallel, then consecutive interior angles are supplementary

Converse of corresponding angle theorem

If corresponding angles are congruent, then lines are parallel

Converse of alternate interior angle theorem

If alternate interior angles are congruent, then lines are parallel

Converse of consecutive interior angle theorem

If consecutive interior angles are supplementary, then lines are parallel

Which one is correct for converse corresponding angle theorem?

If corresponding angles are congruent, then lines are parallel

Line perpendicular to a transversal theorem

If two lines in a plan are perpendicular to the same line, then they are parallel to one another

Perpendicular transversal theorem

If a transversal is perpendicular to one of a pair of parallel lines, then it is perpendicular to the other line

Which one is the perpendicular trasversal theorem?

If a transversal is perpendicular to one of a pair of parallel lines, then it is perpendicular to the other line

Which theorem would be the best for proving that m and n are parallel?

Lines perpendicular to a transversal theorem