Download Journal Chapter 5 Maria Jose Diaz

DESCRIBE WHAT A PERPENDICULAR BISECTOR
IS. EXPLAIN THE PERPENDICULAR BISECTOR
THEOREM AND ITS CONVERSE.
Perpendicular Bisector: is a line perpendicular to a segment at its midpoint.
Perpendicular Bisector Theorem: if a point is one the perpendicular bisector
of a segment, then it is equidistant from the endpoints of the segment.
Converse: if a point is equidistant fro the endpoints of a segment then it is
on the perpendicular bisector of the segment.
EXAMPLES:
Proof:
Statement
Reason
1.L is the perpendicular bisector of AB
Given
2.AM = MB
2. Definition of perpendicular bisector
3.PM is perpendicular to AB
3. Definition of perpendicular bisector
4. angle AMP and angle PMB are right angles
and congruent
4. Definition of perpendicular, any 2 right angles
are congruent.
5. PM = PM
5. Reflexive
6. Triangle AMP congruent Triangle BMP
6. SAS
7. PA = PB
7. CPCTC
DESCRIBE WHAT AN ANGLE BISECTOR IS.
EXPLAIN THE ANGLE BISECTOR THEOREM AND
ITS CONVERSE.
Angle Bisector: a ray that divides and angle into two congruent angles.
Angles Bisector Theorem: if a point is on the bisector of an angle then it is
equidistant from the sides of the angle
Converse: if a point in the interior of an angle is equidistant from the sides
of the angle then it is on the bisector of the angle.
EXAMPLES:
LM
LM=JM
LM= 12.8
DESCRIBE WHAT CONCURRENT MEANS.
EXPLAIN THE CONCURRENCY OF
PERPENDICULAR BISECTORS OF A TRIANGLE
THEOREM. EXPLAIN WHAT A CIRCUMCENTER IS.
Concurrent: when three or more lines intersect at one point.
Concurrency of Perpendicular Bisectors of a Triangle: is a point, a point
where it intersects.
Circumcenter: is the point of concurrency.
EXAMPLES:
DESCRIBE THE CONCURRENCY OF ANGLE
BISECTORS OF A TRIANGLE THEOREM. EXPLAIN
WHAT AN INCENTER IS.
Concurrency of angle bisector: is the point were they intersect in an angle.
Incenter: is the center of the triangle inscribed circle, for example a triangle
has three sides, so then it has three angle bisectors. So the angle
bisectors of a triangle are also concurrent, that is called the incenter of
the triangle.
EXAMPLES:
DESCRIBE WHAT A MEDIAN IS. EXPLAIN WHAT
A CENTROID IS. EXPLAIN THE CONCURRENCY
OF MEDIANS OF A TRIANGLE THEOREM.
Median: a segment whose endpoint are a vertex and the midpoint of the
opposite side.
Concurrency of Medians of a Triangle Theorem: a segment whose endpoint
are a vertex of the triangle and the midpoint of the opposite side.
EXAMPLES:
DESCRIBE WHAT AN ALTITUDE OF A TRIANGLE
IS. EXPLAIN WHAT AN ORTHOCENTER IS.
EXPLAIN THE CONCURRENCY OF ALTITUDES OF
A TRIANGLE THEOREM.
Altitude of a Triangle: a perpendicular segment from a vertex to the line
containing the opposite side.
Concurrency of altitudes of a triangle theorem: if there is a perpendicular
segment in a vertex then it contains a line in the opposite side.
EXAMPLES:
DESCRIBE WHAT A MIDSEGMENT IS. EXPLAIN
THE MIDSEGMENT THEOREM.
Midsegment: a segment that joins the midpoint of two side od the triangle.
Midsegment Theorem: a midsegment of a triangle is parallel to a side of the
triangle and its length is half the length of that side.
EXAMPLES:
DESCRIBE THE RELATIONSHIP BETWEEN THE
LONGER AND SHORTER SIDES OF A TRIANGLE
AND THEIR OPPOSITE ANGLES
The relationship between the longer and shorter sides of a triangle are that
if the legs have the same length or are equal and have the same
measure then the angles opposite have also the same length. As we
know the angles are complementary so then the length of the
hypotenuse is the length of a leg twice.
Angle-Side Relationships:
a. If two sides of a triangles are not congruent, then the larger angle is
opposite the longer side.
b. If two angles of a triangle are not congruent, then the longer side is
opposite the larger angle.
EXAMPLES:
DESCRIBE THE EXTERIOR ANGLE INEQUALITY
Exterior angle inequality: The measure of an exterior angle of a triangle is
greater than the measure of the other interior angle.
EXAMPLES:
DESCRIBE THE TRIANGLE INEQUALITY
Triangle Inequality: the sum of any two sides lengths of a triangle is greater
that the third side length.
EXAMPLES:
DESCRIBE HOW TO WRITE AN INDIRECT PROOF
Indirect Proof: you begin by assuming that the conclusion is false, then you show that
this assumption leads to a contradiction, also called proof of contradiction.
Writing an Indirect Proof:
1. Identify the conjecture to be proven
2. Assume the opposite (the negation) of the conclusion is true
3. Use direct reasoning to show that the assumption leads to a contradiction
4. Conclude that since the assumption is false, the original conjecture must be true.
EXAMPLES:
triangleLMN has at most one right angle.
Step 1: Assume triangleLMN has more
than one right angle. That is, assume
that angle L and angle M are both right
angles. Step 2: If M and N are both right
angles, then m<L = m<M = 90 Step 3:
m<L + m<M + m<N = 180 The sum of
the measures of the angles of a triangle
is 180. Step 4: Substitution gives 90 +
90 + m<N = 180. Step 5: Solving gives
m<N = 0. Step 6: This means that there
is no triangleLMN, which contradicts the
given statement. Step 7: So, the
assumption that <L and <M are both
right angles must be false.
Step 8:
Therefore, triangleLMN has at most one
right angle.
REAL LIFE SITUATIONS:
Sarah left her house at 9:00 AM and arrived at her aunt’s house 80 miles away at
10:00 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed
limit.
Indirect Proof:
Sarah did NOT exceed the 55 mph speed limit.
She drove 80 miles at 55 mph.
At this speed, Sarah would need 80/55 (approximately) = 1 hour 45 minutes to
reach her aunt’s place.
But as per the problem she drove from 9:00 AM to 10:00 AM … exactly an hour.
SO, she must have driven faster than 55 mph….a contradiction that Sarah did NOT
exceed the speed limit.
Therefore, Sarah exceeded the speed limit.
Prove the following using an indirect proof.
if 3n + 1 is even, then ‘n’ is odd.
Indirect Proof:
‘n’ is NOT odd.
‘n’ is even.
Then the given statement is:
if 3n + 1 is even, then ‘n’ is EVEN”
‘n’ is even means ‘n’ is a multiple of 2.
Then:
3n + 1 = 3(2) + 1 = 6 + 1
Well…6 is even. So, 6m + 1 is odd.
Therefore, 3n + 1 is ODD…because 3n + 1 = 6 + 1
By assuming ‘n’ is even, we’ve shown that 3n + 1 is ODD which is a contradiction.
Therefore:
If ‘n’ is odd then 3n + 1 is even. This is the contrapositive of the statement to be
proved.
Since the contrapositive is true, it follows that the original statement "if 3n + 1 is
even, then ‘n’ is odd" is true.
DESCRIBE THE HINGE THEOREM AND ITS
CONVERSE
Hinge Theorem: if two sides of one triangle are congruent to two sides of
another triangle and the included angles are not congruent, that the
longer third side is across from the larger included angle.
Converse: if two sides of one triangle are congruent to two sided of another
triangle and the third sides are not congruent, then the larger included
angle is across the longer third side.
EXAMPLES: