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Geometry, Chapter 5
Section 5.1: Types of angles created by two lines cut by a transversal. Properties of angles created by a
transversal cutting two parallel lines.
1. Transversal – a line that intersects two distinct lines in two distinct points.
l
l
Transversal
Not a Transversal
l
2. Angles Formed by a Transversal
x
Alternate Interior Angles - ∠ w and ∠ r, ∠ z and ∠ s
y
z w
r
Alternate Exterior Angles - ∠ y and ∠ t, ∠ x and ∠ v
Corresponding Angles - ∠ x and ∠ r, ∠ z and ∠ t, ∠ y and ∠ s, ∠ w and ∠ v
m
t
s
v
Cointerior Angles (or Same Side Interior) - ∠ w and ∠ s, ∠ z and ∠ r
Coexterior Angles (or Same Side Exterior) - ∠ x and ∠ t, ∠ y and ∠ v
3. Exterior Angle Theorem: In a triangle, an exterior angle is greater than either nonadjacent interior angle.
sketch:
4. Euclidean Parallel Postulate: Given a line l and a point P, not on l, there can be only one line parallel to l
through the point P.
∙
P
l
5. Theorem: Two lines are cut by a transversal are parallel if and only if a pair of alternate interior angles are
congruent.
sketch:
6.
Corollary: Two lines cut by a transversal are parallel if and only if a pair of corresponding angles are
congruent.
sketch:
7.
Corollary: Two lines are cut by a transversal are parallel if and only if a pair of same side interior angles
are supplementary.
sketch:
8.
Corollary: Two lines are perpendicular to a third line if and only if they are parallel.
sketch:
n
Geometry, Chapter 5
9. Indirect Proof or Proof by Contradiction
A theorem usually consists of two parts: the hypotheses (given statements) and a conclusion (to prove.)
Method of Direct Proof: We start by assuming all of the hypotheses are true and then produce a sequence of
statements, each of which follows logically from previous statements, the hypotheses, postulates, definitions,
or other proven theorems. The final statement should be the conclusion of the theorem.
Method of Indirect Proof: We assume the hypotheses are true as before, but in addition we assume that the
conclusion of the theorem is false. Our goal is then to produce a contradiction which is caused by these
assumptions so that it is clear that whenever the hypotheses are true it is impossible for the conclusion to be
false. Therefore, whenever the hypotheses are true the conclusion must also be true, which proves the
theorem.
Example of an Indirect Proof:
Theorem: If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then
the lines are parallel.
(Note: This is one half of the “if and only if” theorem from number 5 above. Our proof will also use
the Exterior Angle Theorem from number 3 above.)
Proof: Let’s identify the hypotheses and the conclusion of this theorem.
Hypotheses: Two lines are cut by a transversal and a pair of alternate interior angles are congruent.
Conclusion: The two lines are parallel.
We begin by assuming the hypotheses are true, which we can show using the following diagram:
m
1
(note: we don’t know that l is parallel to m yet)
2
l
t
Next we assume the conclusion is false, that the lines are not parallel and so must intersect somewhere:
1
2
3
l
m
t
This immediately contradicts the Exterior Angle Theorem because one of the angles becomes an
exterior angle to the triangle formed when the lines intersect, let’s say its ∠1 as shown. Then the
Exterior Angle Theorem requires that ∠1 > ∠2 , which is incompatible with our hypotheses that
∠1 =∠2 . So we conclude that whenever a pair of alternate interior angles are congruent the two lines
cannot intersect and therefore must be parallel.
Geometry, Chapter 5
Section 5.2: Sum of the angles in a triangle. Related theorems.
1. Theorem: The sum of the angles of a triangle is 180°.
2.
Corollary: (Ext Angle Theorem 2) An exterior angle of a triangle equals the sum of the two nonadjacent
interior angles.
3.
Corollary: (AAS Congruence) If two angles and a side of one triangle are congruent respectively to two
angles and a side of another triangle, then the two triangles are congruent.
4.
Corollary: (HA Congruence) If the hypotenuse and one acute angle of a right triangle are congruent
respectively to the hypotenuse and one acute angle of another right triangle, then the triangles are
congruent.
5.
Angle Bisector Theorem: A point is on the bisector of an angle if and only if it is equidistant from the
sides of the angle.
Section 5.3: Properties of parallelogram and rhombus.
1. Parallelogram
Definition: A parallelogram is a quadrilateral whose opposite sides are parallel.
B
Theorem:
A diagonal divides the parallelogram into two congruent triangles.
Corollary:
Corollary:
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Theorem:
Theorem:
The diagonals of a parallelogram bisect each other.
Consecutive angles of a parallelogram are supplementary.
Theorem:
Parallel lines are the same distance apart everywhere.
Definition:
The distance between a point and a line is the length of the segment from the point
perpendicular to the line.
A
C
D
B
A
C
D
2. Proving a quadrilateral is a parallelogram
Prove that both pairs of opposite sides are parallel. (Definition)
Theorem:
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem:
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem:
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the
quadrilateral is a parallelogram.
Theorem:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
3. Rhombus
Definition:
Theorem:
Theorem:
Theorem:
Theorem:
Theorem:
A rhombus is a quadrilateral with all sides congruent.
A rhombus is a parallelogram.
The diagonals of a rhombus are perpendicular to each other.
The diagonals of a rhombus bisect the angles.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If the diagonals of a parallelogram bisect the angles, then the parallelogram is a rhombus.