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Geometry
Review
I.
Congruent Figures.
A. Things to know.
1. Naming parts of congruent figures from figures and a name.
2. Solving for x and finding the measure of angles and length of
sides of congruent figures.
V. Isosceles and Equilateral Triangles.
A. Base angles of an isosceles triangle are congruent.
1. The converse is true.
a. two angles of a triangle are congruent then the opposite
sides are congruent.
i. hence the triangle is isosceles.
2. Equilateral triangles are equiangular.
a. the converse is true.
B. The angle bisector of the vertex angle of an isosceles triangle is a
perpendicular bisector.
C. Finding the measure of angles and length of sides of an isosceles
triangle.
D. Things to know.
1. Finding for missing angle and segment lengths in an isosceles
and equilateral triangle.
2. Finding x in order find missing angles and segment lengths in
an isosceles and equilateral triangle.
I.
Midsegments of Triangles. (Chapter 5)
A. Parallel to the third side of the triangle.
B. Equal to ½ the length of the third side of the triangle.
C. Things to know.
1. Find the length of midsegments and sides of a triangle.
2. Find the measure of angles formed by the midsegments.
3. Using the midsegment to find the length of objects.
4. Solving for x in an expression using the properties of
midsegments.
II.
Perpendicular and Angle Bisectors in Triangles.
A. Perpendicular bisectors.
1. Points on the perpendicular bisector is equidistant to the
endpoints of a segment.
2. Points equidistant to the endpoints of a segment lie on the
perpendicular bisector of that segment.
B. Angle Bisectors.
1. Points on the angle bisector is equidistant to the sides of the
angle.
2. Points equidistant to the sides of an angle lie on the angle
bisector.
C. The shortest distance from a point to a line is the perpendicular
distance from the point to the line.
D. Things to know.
1. Identify or sketch a perpendicular or angle bisector.
2. Find the length of segments using the properties of
perpendicular and angle bisectors.
III. Bisectors in Triangles.
A. Concurrent lines and points of concurrency.
1. Perpendicular bisectors = Circumcenter.
a. Circumcenter- point of concurrency of the perpendicular
bisectors.
1a. equidistant to the vertices of a triangle or three
points.
2a. you need to find this point to find the center of the
circle that will circumscribes the triangle.
i. all triangles can be circumscribed, it’s the only
polygon like this.
2. Angle Bisectors = Incenter.
a. Incenter- point of concurrency of the angle bisectors.
1a. equidistant to the sides of a triangle.
2a. you need to find this point to find the center of the
circle that will be inscribed in the triangle.
B. Things to know.
1. Know which lines form what points of concurrency.
2. Know the properties of the points of concurrency.
3. Be able to locate these points by graphing or by sketching and
labeling diagrams.
4. Finding x and length of segments and measure of angles.
IV. Medians, and Altitudes and their points of concurrency.
A. Median.
1. A segment that joins the vertex of a triangle to the midpoint of
the opposite side.
B. Altitude.
2. A segment that joins the vertex and is perpendicular to the
line that contains the opposite side.
C. Know the two points of concurrency, the lines that form them, and
their properties.
1. Centroid- point of concurrency of the medians of a triangle.
a. the centroid divides the median such that the length from
the centroid to the midpoint is half as long as the length
from the centroid to the vertex.
b. the distance from centroid to the midpoint is 1/3 the
length of the median.
c. the distance from the centroid to the vertex is 2/3 the
length of the median.
d. center of gravity of a triangle.
2. Orthocenter- point of concurrency of the altitudes of a
triangle.
D. Things to know.
1. Know which lines form what points of concurrency.
2. Know the properties of the points of concurrency
3. Identify a segment as a median or an altitude for all triangles.
4. Use the centroid to find the lengths of segments.
5. Draw and label medians, altitudes and how to find points of
concurrency.
VI. Inequalities in Triangles.
A. The largest side is opposite the largest angle and the smallest side
is opposite the smallest angle.
B. The sum of any two sides of a triangle must be larger than the third
side of the triangle.
C. Things to know. (Pg.276-2771-27)
1. Given the angles of a triangle order the sides from least to
greatest in length.
2. Given the lengths of the sides of a triangle order the angles
from least to greatest in measure.
3. Given three possible lengths for a triangle can a triangle be
formed.
4.
Given two sides of a triangle find all possible lengths of the
third side.
VII. Inequalities in Two Triangles.
A. Hinge Theorem and its converse.
B. Things to know.
1. Relating side lengths of two different triangles.
2. Finding a range of possible values for x in a given situation.