Chapter 4 Review -- Lines in the Plane

4.1 -- Detours and Midpoints

A detour proof is a proof that involves proving more than one pair of triangles congruent to solve the problem.

Procedure for Detour Proofs:

1. Determine which triangles you must prove to be congruent to reach the required conclusion.

2. Attempt to prove the triangles congruent. If you can't, take a detour. (See steps 3-5 below)

3. Establish which parts of the diagram you must prove to be congruent to prove the congruence of the triangles.

4. Find a pair of triangles that you can prove congruent and contain a pair of parts need for the main proof. (See step 3)

5. Prove triangles in step 4 congruent.

6. Use CPCTC and complete proof.

Midpoints sometimes need to be located in geometry problems. They can be determined by the following formula :

4.2 -- Case of the Missing Diagram

Sometimes geometry problems will not have a diagram but you will be given a statement and are expected to set up your own problem. It is important that you can effectively set up a problem correctly. There are several techniques you can use.

"If...then..." -- The given information is usually found in the "if" clause and what we need to prove is usually in the "then" clause.

Example: If the base of an isosceles triangle is extended in both directions,

then the exterior angles formed are congruent.

Given:△ABC Iso.

Prove:∠ABX≅∠ACY

If the word "then" is left out, or the conclusion comes before the hypothesis, the hypothesis is always follows the word "if" and always contains the givens.

Example: If a triangle is isosceles, the triangle formed by its base and the angle bisectors of its base angles is also isosceles.

Given:△ABC iso. Base BC

CX bis ∠ACB

BX bis∠ABC

Prove:△BXC iso.

If "if" and "then" are left out, this is a hint that the statement begins with the given information and ends with the conclusion.

Example: The median to the base of an isosceles triangle divides the triangle into two congruent triangles.

Given:△XYZ iso. Base YZ

XW median to YZ

Prove:△XYW≅△XZW

4.3 -- A Right-Angle Theorem

Theorem -- If two angles are both supplementary and congruent, they are right angles.

Example:

In the proof above, the use of the theorem is highlighted.

4.4 -- The Equidistance Theorems

The distance between an object is the length of the shortest path joining them.

Postulate:

A line segment is the shortest path between two points.

If two points are the same distance from a third point, the third point is said to be equidistant from the first two points.

The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.

-- In an isosceles triangle, the median is the altitude, and thus implies that it is the perpendicular bisector.

Theorems:

If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment.

On isosceles triangle LMN, if LP is the perpendicular bisector, we can determine that MP is congruent to PN because perpendicular bisector implies congruent segments. We can also imply that angle LPM and angle LPN are right angles because perpendicular bisector implies right angles.

4.5 -- Introduction to Parallel Lines

A plane is a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface.

A plane only has length and width, both infinite. A plane has no thickness.

If points, lines, etc. lie on the same plane, they are said to be coplanar. If points, lines, etc. don't lie on the same plane, they are noncoplanar.

Transversals

In the figure above, AB is the transversal of lines EF and CD. A transversal is a line that intersects two coplanar lines in two distinct points.

The area shaded green is considered the interior of the figure. The area shaded blue is considered the exterior region.

In the figure, angles 1 and 5 are corresponding angles. Look for an F shape to find corresponding angles. Angles 3 and 6 are alternate interior angles. Look for an N or Z shape. Angles 2 and 7 are alternate exterior angles.

Parallel lines are two coplanar lines that do not touch.

4.6 -- Slope

The slope (m) of a nonvertical line, segment, or ray containing (x1, y1) and (y1, y2) is defined by the following formula:

It doesn't matter which point is chosen as X1 or Y1.

When the slope formula is used on a vertical line, the denominator is zero. Since division by zero is undefined, the line has an undefined slope. When the formula is used on a horizontal line, the slope is zero.

Summary

Rising line --> positive slope

Horizontal line --> zero slope

Falling line --> negative slope

Vertical line --> undefined slope

Slopes of Parallel and Perpendicular Lines

Theorem: If two nonvertical lines are parallel, they have equal slopes.

The converse of that theorem is also true. (If the slopes of two nonvertical lines are equal, then the lines are parallel.)

Theorem: If two lines are perpendicular and neither is vertical, each line's slope is the opposite reciprocal of the other's.

The converse is also true. (If a line's slope is the opposite reciprocal of another line's slope, the two lines are perpendicular.)

Sorry for this ridiculously long blog post! Hope it helped.

-Olivia Sheridan