Download Geometry Final Exam Review

Geometry Final Exam Review
1. Triangles



In an equilateral triangle, all sides are of equal length.
In an isosceles triangle, at least two sides are of equal length. An isosceles triangle also
has two congruent angles (namely, the angles opposite the congruent sides).
In a scalene triangle, all sides have different lengths. The internal angles in a scalene
triangle are all different.[3]
Equilateral



Scalene
A right triangle has one 90° angle. The side opposite to the right angle is the
hypotenuse; it is the longest side in the right triangle. The other two sides are the legs of
the triangle.
An obtuse triangle has one angle larger than 90° (an obtuse angle).
An acute triangle has angles that are all smaller than 90°
Right
One 90 degree angle
•
Isosceles
Obtuse
Acute
One angle larger than 90 degrees
All angles are smaller than 90 degrees
The measure of the angles of a triangle must equal 180 degrees.
A + B + C = 180
• When asked to identify the shortest or longest side of a triangle, you must find the
smallest or largest angle and move across to find the side.
Shortest side
If one angle of a triangle has a greater
measure than another angle, then the side
opposite the greater angle is longer than the
side opposite the lesser angle.
Longest side
When asked to identify the largest or smallest angle of a triangle, you must for the longest or
shortest side and move across:
Smallest angle
If one side of a triangle is longer than another
side, then the angle opposite the longer side
has a greater measure than the angle opposite
the shorter side.
Largest angle
2. Triangle Inequality Theorem- The sum of the lengths of any two sides of a triangle is
greater than the length of the third.
Hint - For example would sides of length 4, 5 and 6 form a triangle....?

4 + 5 > 6 , 4 + 6 > 5, 5 + 6 > 4 Yes, it’s a triangle!
How about sides of length 4, 11, and 7

4 + 11 > 7, 11 + 7 > 4, 4 + 7 = 11, since 4 + 7 is not > than 11, it is not a triangle!!
An observant student mentioned last year that all you have to do is add the two smallest sides
to see if it is larger than the other.... How’s that for easy???
5. Parallelograms:
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles in a parallelogram
are supplementary.
The diagonals of a parallelogram bisect each other
Parallelogram Property Summary
1.
2.
3.
4.
5.
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
Diagonals bisect each other.
A pair of opposite sides is both parallel and congruent
Rectangles:
A rectangle is a quadrilateral with four right angles.
If a parallelogram is a rectangle, then its diagonals are congruent
Rectangle Property Summary
1.
2.
3.
4.
5.
6.
Opposite sides are congruent and parallel. (All parallelograms)
Opposite angles are congruent. (All parallelograms)
Consecutive angles are supplementary. (All parallelograms)
Diagonals bisect each other. (All parallelograms)
Diagonals are congruent. (Rectangle)
All four angles are right angles. (Rectangle)
Squares:
A square is a quadrilateral with four right angles and four congruent sides.
Rhombus:
A Rhombus is a quadrilateral with four congruent sides.
The diagonals of a rhombus are perpendicular.
(Hints): That makes all four angles where the diagonals intersect 90 degrees.
If the diagonals of a parallelogram are perpendicular,
then the parallelogram is a rhombus.
Each diagonal of a rhombus bisects a pair of opposite angles.
---(Hints): If you look carefully you should notice that the four triangles formed by the rhombus's
diagonals are congruent.
Trapezoids
A trapezoid is a quadrilateral with at least one pair of parallel sides.
While this shape looks quite different... it is actually quite a common shape. The parallel sides
are called BASES. The nonparallel sides are called LEGS. There are two pairs of base angles,
the two touching the top base,
and the two touching the bottom base.
ISOSCELES TRAPEZOID - If the legs of a trapezoid are
congruent, then the trapezoid is an isosceles trapezoid.
**Both pairs of base angles of an isosceles trapezoid are
congruent.
**The diagonals of an isosceles trapezoid are congruent.
The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the
measures of the bases. In other words, add the two bases together and divide by two and you’ll
have the measure of the median.
If you’re given the measures of one base and the median and asked to find the measure of the
other base, then use the formula b1 + b2 = 2 (median). In other words the base you know plus the
missing base is equal to two times the median.
Proportion:
You can use ratios to solve problems. A RATIO is a comparison of two quantities. The ratio of
a to b can be expressed as , where b is not zero. The ratio can also be written as a:b. For
example in a class of 10 girls and 15 boys we could express this as a ratio of girls to boys, 10:15.
If we wanted to express those same numbers in a ratio of boys to girls, it would be 15:10. Often
a ratio can be easily express in a reduced form... for instance instead of 10:15, we could reduce it
to 1:1.5 or 2:3.
An equation stating that two ratios are equal is a PROPORTION. For example:
implies that the ratio of a:b is equal to c:d. A simple example would be
This
.
Similar Polygons:
Definition of Similar Polygons - Two polygons are SIMILAR if and only if their corresponding
angles are congruent and the
measures of their corresponding
sides are proportional.
The ratio of lengths of two corresponding sides of two similar polygons is called the SCALE
FACTOR. The scale factor provides the ratio of all sides of two similar shapes. To find the
scale factor set up your corresponding sides like the one above. Then fill in all the numbers you
know. Find the one that has a number on the top and bottom, that’s your scale factor. If you can
reduce it, you should.
Circles:
A CIRCLE is the set of all points in the plane that are a given distance from a given point. The
given point is called the CENTER of the circle. A RADIUS of a circle is a segment that has one
endpoint at the center of the circle and the other endpoint on the circle.
A circle is usually named by its center. The circle to the left has a center point
of A, so the circle is called circle A
Chords - A CHORD of a circle is a segment that has its endpoints on the circle.
Chord CD, chord GH, and chord EF.
Diameters - A DIAMETER of a circle is a chord that contains the center of the
circle. Diameter CV, diameter DT, diameter GH, diameter EU, etc.....
Radii - A RADIUS of a circle is a segment that has one endpoint at the center
of the circle and the other endpoint on the circle. Radius CA, radius EA, radius
HA, etc..
Circumference - The CIRCUMFERENCE (perimeter) is the distance around the circle. To
calculate the circumference of a circle you multiply pi times the diameter or pi times twice the
radius. (C = 2 r or C = d, r = radius and d = diameter)
AREA:
is the calculation of the number of square units in the interior of a 2-dimensional object. The
most basic shapes to calculate area on are squares, rectangles and parallelograms. The formula is
Area = base x height.
Shapes may also not fit the given formulas - when this happens break the shape down into
smaller shapes that you know area formulas for.
A triangle is always exactly half of a rectangle. No matter
what shape a triangle takes on - it will always have a
rectangle surrounding it that will be twice its area. In the
provided diagram you can see that in the triangle there is
Area 1 + Area 2 and in the rectangle that circumscribes it
there is two Area 1's and two Area 2's. This relationship
provides us with the triangle area formula being one-half
the rectangle formula.
The Trapezoid and Rhomus Area Formulas are a little less obvious as the triangle formula but
they too have explanations. For now this site will simply provide you with the formulas and let
your teacher demonstrate their origin.