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Transcript
Chapter 5: Congruence based on Triangles
Definitions:
Line Segments Associated with Triangles
Altitude of a
A line segment drawn from any vertex of the triangle,
Triangle
perpendicular to and ending at the opposite side.
Median of a
A line segment that joins any vertex of the triangle to the
Triangle
midpoint of the opposite side.
Angle Bisector A line segment that bisects any angle of the triangle and
of a Triangle
ends in the side opposite that angle.
Perpendicular
A line or segment that is perpendicular to another segment at
Bisector
its midpoint.
Circumcenter
The point where the three perpendicular bisectors of a
triangle intersect.
Triangles
Isosceles
A triangle with at least two sides congruent.
Triangle
Equilateral
A triangle with three sides congruent.
Triangle
Postulates, Theorems and Corollaries
Isosceles If two sides of a triangle are congruent, the angles opposite those
Triangle
sides are congruent.
Theorem
OR
The base angles of an isosceles triangle are congruent.
Corollary
Corollary
Corollary
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
The median from the vertex angle of an isosceles triangle bisects
the vertex angle.
The median from the vertex angle of an isosceles triangle is
perpendicular to the base. (altitude)
Every Equilateral Triangle is Equiangular.
** An Equilateral Triangle has all the properties of an Isosceles
Triangle.
If two points are each equidistant from the endpoints of a line
segment, then the points determine the perpendicular bisector of
the line segment.
If a point is equidistant from the endpoints of a line segment,
then it is on the perpendicular bisector of the line segment.
If a point is on the perpendicular bisector of a line segment, then
it is equidistant from the endpoints of the line segment.
A point is on the perpendicular bisector of a line segment if and
only if it is equidistant from the endpoints of the line segment.
The perpendicular bisectors of the sides of a triangle are
concurrent.
The perpendicular bisectors of a triangle intersect at a point that
is equidistant from the vertices of the triangle.
Radii of congruent circles are congruent.
All radii of the same circle are congruent
TIPS for Overlapping Triangles and Using 2 pair of Congruent Triangles:
1.
2.
3.
4.
Draw good diagrams – make sure they are large enough and labeled.
Mark the given(s) on the diagram (it’s the law!)
Be neat and orderly.
For Overlapping Triangles:
a. Be sure to identify the correct triangles needed to prove congruent.
b. You may need to prove a 1st pair of triangles congruent in order to prove a
2nd pair of triangles congruent.
c. Sometimes it may be easier to separate the triangles or use different
colors (or solid & dotted lines) to differentiate each triangle.
TIPS for proving intersecting lines or segments are perpendicular:



The lines intersect to form right angles.
(defn of perpendicular lines)
The lines intersect to form congruent adjacent angles
(theorem – if two lines intersect to form congruent adjacent angles, then they
are perpendicular).
One line contains 2 points that are equidistant from the endpoints of a
segment of the other line.
(theorem - If a point is equidistant from the endpoints of a line segment, then
it is on the perpendicular bisector of the line segment)