Written and Compiled by the ProcrastiNote Team
Last Updated: January 16, 2021
Contents
1 Vocabulary 2
2 Properties of Parallel Lines 2
3 Parallel Line Relationships 2
3.1 Perpendicular Lines . . . 2 3.2 Triangles . . . 3 4 Slope 3 4.1 Parallel Lines . . . 3 4.2 Perpendicular Lines . . . 3 5 Construction 4 5.1 Parallel Lines . . . 4 5.2 Perpendicular Lines . . . 4
6 Spherical and Euclidean Geometry 5
A Credits 6
A.1 Contributions . . . 6 A.2 External Sources . . . 6 A.3 Image Credits . . . 6
1
Vocabulary
• Alternate interior anglesare congruent angles located on opposite sides of the innermost part of the transver-sal.
• Alternate exterior anglesare congruent angles located on opposite sides of the outermost part of the transver-sal.
• Corresponding angleshave the same position on the same side of the transversal. • Same side interior anglesare in the same innermost side of the transversal. • Same side exterior anglesare in the same outermost side of the transversal. • Transversalsare lines that intersect more than two lines at a time.
2
Properties of Parallel Lines
Parallel lines have corresponding angles, alternate interior/exterior angles, and same side interior/exterior an-gles.
Postulates and Theorems
• Same - Side Interior Angles Postulate:"If a transversal intersects two parallel lines, then the same-side interior angles are supplementary."
• Alternate Interior Angles Theorem:"If a transversal intersects two parallel lines, then alternate inte-rior angles are congruent."
• Corresponding Angles Theorem:"If a transversal intersects two parallel lines, then corresponding angles are congruent."
• Alternate exterior Angles Theorem:"If a transversal intersect two parallel lines, then alternate exte-rior angles are congruent."
3
Parallel Line Relationships
If two lines are parallel to the same given line, then they are proven to be parallel to each other.
How Many Perpendicular Lines?
A perpendicular line is an intersection that forms a right (90 degree) angle between both lines. Although many intersections can be made on a line, there can only be one perpendicular line in each given line.
3.2
Triangles
1. Triangle Angle Sum Theorem:All of a triangle’s angles summed up equals 180 degrees.
∠A+∠B+∠C= 180 (1)
2. Exterior angles of a polygonare formed by an extension of a side adjacent of the second side.
3. Remote interior anglesare used to find the measure of each exterior angle since they are nonadjacent to the angle.
4. Triangle Exterior Angle Theorem:The measure of a triangle’s exterior angles can be found with the sum of two interior remote angles.
4
Slope
Slope is the rate of change in x (run) and y (rise) values. There are positive, negative, zero, and undefined slopes in coordinate planes. Slope-intercept form is expressed asy = mx+bwheremis the slope,xis the variable,
andbis the y-intercept. Point-slope form is shown asy−y1 =m(x−x1)wheremis the slope and(x1, y1)is an
ordered pair found on the given line.
4.1
Parallel Lines
If two lines line 1 and line 2 have slopesm1andm2respectively, and line 1 is line 2, thenm1equalsm2. If two
lines are parallel, then their slopes are equal. The converse of the previous statement is also true. Did you figure it out? If their slopes are equal, the lines are parallel.
4.2
Perpendicular Lines
If two lines line 1 and line 2 are perpendicular andm1andm2are their slopes respectively, then the product of
their slope is -1. Converse of previous statement: If the product of their slope is -1, then the lines are perpendic-ular. When two lines are perpendicular, their slopes are the reciprocal of each other.
5
Construction
5.1
Parallel Lines
Construct a parallel line to the given line using a point NOT on the line.
1. Draw a point on the given line and label it as A. Connect it with the point not on the line with a straight-edge.
2. Open the compass slightly longer than half the distance between points A and the given point, then draw an arc that intersects both lines to create corresponding angles.
3. With the same radius, place the compass on the given point and create an arc.
4. Measure the distance of the arc between the intersections on point A. With the same measured distance, place it on the given point and draw an arc that intersects with the previous one.
5. Label the arc intersection as point B and connect it with the given point using a straightedge.
5.2
Perpendicular Lines
Construct a line perpendicular through a point on a line.
1. Set the compass on the given point and make an arc on the right side of the line. 2. With the same radius, create an arc on the left side of line.
3. Label the points as A and B.
4. Place the compass on point B and open the radius slightly more than half the previous radius. 5. Repeat for point A so that the arcs intersect.
6. Label the intersection as point C and connect it with point P with a straightedge so that it makes a 90 de-gree angle.
Construct a perpendicular line through a point NOT on the given line.
1. Place the compass on point P and draw two arcs (left and right side) that intersect the line. Label the points as A and B.
2. Set the compass on point B (right side) and draw an arc with the same radius. Repeat for point A on the left side.
6
Spherical and Euclidean Geometry
Spherical geometryis the observance of a sphere’s surface. The theory states that the surface will result in a great circle, commonly known as a plane, when cut into a hemisphere. Postulate: If there is no point on the line, then no line is parallel to the given line. Why? Because, in spherical geometry, lines are great circles that always intersect with each other at two points. Never is a line parallel to each other since they are perpendicular. The-orem: Unlike in Euclidean geometry, the sum of a triangle’s angles is always greater than 180 degrees. Why? Be-cause they triangle is placed on three points inside a great circle and the sides of the triangle are curved, result-ing in modified angles.Euclidean geometryis the study of lines and anything that contains them like planes and points.
Appendices
A
Credits
A.1
Contributions
• Drafted byEmily Ramirez
• Contributions by:
– Austin Wang
A.2
External Sources
• Pearson Texas Geometry Student Text and Homework Helper
A.3
Image Credits
• Clock: Veronica Cruz
B
Extra Resources
• All resources mentioned retain the same licensing as the original creators intended. The ProcrastiNote Team has checked every single one of these to make sure that our usage of their materials remains within the legal realm. If you are the owner of these materials and you believe there is a mistake in our citations, please contact us [email protected]. Thanks!
