Chapter 4
4.1 Detours and Midpoints
Sometimes when you attempt to solve a proof the pair of triangles that you
need can't be proven congruent until another set are proven congruent. Once
the first set are congruent their corresponding parts allow you to prove the
second set congruent. These proofs are referred to as DETOUR proofs because
you have to take the long way around to get to your solution.
Example:
Midpoints- Formulas
Given a number line, if you need to find the coordinate of the midpoint of the
segment joining two point - average the coordinates of the segment's
endpoints.
Example:
If the coordinate of E is -18 and the coordinate of F is 12, find the coordinate of
the midpoint of EF.
(-18 + 12)/2 = 6/2 = 3
Given a coordinate grid and a segment joining two points on the plane, to find
the coordinates of the midpoint of the segment, average the x coordinates and
average the y coordinates.
Formula :
4.2 The Case of the Missing Diagram
Many problems in geometry are missing diagrams to help you solve a proof.
When this occurs you must be able to set up the picture and define your given
and prove statements.
Example: Prove that the median to the base of an isosceles triangle is also an
altitude.
Note that the diagram needs to be isosceles with a median drawn to the base
from the vertex angle.
In the following the diagram is missing as well as the formal statement of the given and prove.
Try drawing the diagram before checking your solution for each.
1. The segment joining the midpoints of the sides of a rectangle form a quadrilateral with 4
congruent sides.(Rhombus)
2. If segments are drawn from the midpoints of the legs of an isosceles triangle, perpendicular to
the base, then the segments are congruent.
3. The midpoint of the hypotenuse of a right triangle is equidistant from the 3 vertices of the
triangle.
4. If two triangles are congruent then the corresponding altitudes are congruent.
1. answer
2. answer
3. Answer
4. Answer
4.3 A right angle theorem
When you are asked to prove that segments are perpendicular, you need to
prove that the angles are right angles. When angles are adjacent and congruent
it can be proven that each must be a right angle. Rather than having to go
through all the steps each time, we use the theorem below.
CSAR
If two angles are both congruent and supplementary, then they are right angles.
Proof:
F
Given: 1  2
1 and 2 are supplements
Prove: 1 and 2 are Right angles
E
2
1
H
G
See if you can write the reasons for the following proof.
C
Given: D is the midpoint of AB
ABC is Isosceles with base BA
Prove: CD  AB
B
D
A
1. D is the midpoint of AB
2. BD  AD
3. ABC is Isosceles with base BA
4. BC  AC
5. CD  CD
6. CDB  CDA
7. CDB  CDA
8. CDB and CDA are supplements
9. CDB and CDA are right
10. CD  AB
( 1. Given 2. Def of midpoint 3. Given 4. Def of Isos 5. Reflexive 6) SSS 7.
CPCTC 8. L.P. Sup 9. CSAR 10. Def of Perpendicular)
4.4 Equidistance Theorems
Distance between two points is defined as the length of the segment joining
those two points.
Distance between a point and a line is the perpendicular to the line.
You would not choose a diagonal route from point F to line DE as the length
would be greater. The shortest distance between a point and a line is the
perpendicular to the line.
Equidistant - means the distance is the same.
The first theorem we will refer to as the PBT - or perpendicular bisector
theorem.
If a point lies on the perpendicular bisector of a segment then the point is
equidistant from the endpoints of the segment.
Although you can still prove the segments congruent the long way, this will
shorten some of our previous proof. Below are the steps using the PBT and
using congruent triangles. Note the shortened length.
CPBT - Converse of the Perpendicular Bisector Theorem.
If two points are equidistant from the endpoints of a segment, then the line
containing those points is the perpendicular bisector of the segment.
Example of proof using CPBT
\
4.5 Introduction to parallel lines
Parallel Lines are two coplanar lines that never intersect.
The symbol for parallel lines is . AB CD
Skew lines are non coplanar lines that do not intersect.
Z
X
Y
XY and ZW are skew lines
W
Transversal is a line that intersects two or more coplanar lines in distinct points.
The lines need not be parallel.
When a transversal intersects two lines 8 angles are formed. Pairs of these
angles have special names according to their positions.
Corresponding angles are angles located in the same position at the
intersections of the transversal with the two lines.
1 & 5, 2 & 6, 3 & 8, 4 & 7 are the pairs of corresponding s.
Interior angles are the angles formed between the two intersected lines.
4, 3, 5, & 6 are interior angles.
Alternate interior angles are non-adjacent angles formed on opposite sides of
the transversal in the interior.
4 & 6, 3 & 5are alt. int. s
Same-side interior angles are non adjacent angles in the interior that are on the
same side of the transversal.
3 & 6, 4 & 5 are same  side int. s
Alternate exterior angles are non-adjacent angles formed on opposite sides of
the transversal in the exterior.
1 & 8, 2 & 7are alt. ext. s
Same-side exterior angles are non adjacent angles in the exterior that are on the
same side of the transversal.
1 & 7, 2 & 8are same side ext. s
Skew lines are lines are non-coplanar lines that do not intersect.
4.6 Slopes of Lines
We will be proving some of our theorems in the future using coordinate
proofs. This requires knowing the Cartesian Coordinate Grid and some of the
formulas and rules applying to points and lines. The midpoint formula from
section 4.4 will be of use to us as well as the slope formula which you have
already learned in Algebra but we will repeat here.
When lines are drawn on a plane, their slant is referred to as the slope of the
line.
A number is assigned to slope and is determined by choosing any two points on
the line and finding the change in their y coordinates and dividing this by the
change in the x coordinates.
Formula for slope(m):
Lines that slope up to the right have a positive slope.
Lines that slope down to the right have negative slopes
Horizontal lines have zero slope
Vertical lines have undefined slope.
Three tests of position of lines and points on a plane.
Collinearity - To test if three points are collinear, find the slopes of the lines
between each set of points and if it is the same, then the points are collinear.
Example:
Slope of CD = 1/2
Slope ED = 1/2
Slope of EC = 1/2 therefore the lines are collinear
To test if lines are parallel, make sure they have the same slope.
Parallel Lines have the same slope
To test if lines are perpendicular, check to see if their slopes are negative
reciprocals. (Easy way to check is to multiply the slopes and if the solution is
negative one, they are negative reciprocals of each other).
Perpendicular Lines have negative reciprocal slopes. (non-vertical or
horizontal)
If two lines are vertical, they are parallel
If two lines are horizontal, they are parallel.
In a plane, a vertical line and horizontal line are perpendicular.
Example:
Given the diagram with triangle ABC. Find the slope of the altitude to BC,
Find the length of the median to BC and find the slope of AD if it is parallel to
BC.
Slope of BC = 8/12 or 2/3
Slope of AN (altitude) = -3/2 (negative reciprocal)
The coordinates of M are (11,9) so the Slope of AM = -6/8 or - 3/4
The slope of AD // BC = 2/3 since it must be the same.