Download Angle Bisector Theorem

Chapter 5 Relationships
within Triangles
 Midsegments
 Perpendicular bisectors - Circumcenter
 Angle Bisectors – Incenter
 Medians – Centroid
 Altitudes – Orthocenter
 Inequalities in one triangle
 Inequalities in Two Triangles
Midsegment
Finding Lengths
Perpendicular Bisector
Theorem
 If a point is on the perpendicular bisector of a
segment, then it is equidistant from the endpoints of
the segment
Converse of the Perpendicular
Bisector Theorem
 If a point is equidistant from the endpoints of a
segment, then it is on the perpendicular bisector of
the segment
Using the Perpendicular
Bisector Theorem
 What is the length of QR?
 How would you set up the problem?
Angle Bisector Theorem
 If a point is on the bisector of an angle, then the
point is equidistant from the sides of the angle
Converse of the Angle
Bisector Theorem
 If a point in the interior of an angle is equidistant
from the sides of the angle, then the point is on the
angle bisector.
Concurrency of Perpendicular
Bisectors Theorem
 The perpendicular bisectors of the sides of a triangle
are concurrent at a point equidistant from the
vertices
Concurrency of Angle
Bisectors Theorem
 The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides of
the triangle
Concurrency of Medians
Theorem
Altitude of a Triangle
 The perpendicular segment from the vertex
of the triangle to the line containing the
opposite side
 Can be on the inside, the outside, or a side of
a triangle
Summary
Corollary to the Triangle
Exterior Angle Theorem
 The measure of an exterior angle is greater than the
measure of each remote interior angles of a triangle
Applying the Corollary
Theorem
 If two sides of a triangle are not congruent, then the
larger angle lies opposite the longer side
Theorem
 If two angles of a triangle are not congruent, then
the longer side lies opposite the larger angle
Take Note
 In order to form or construct a triangle the sum of
the two shortest sides must be greater than the
largest side.
Triangle Inequality
Theorem
Find the Possible Lengths
The Hinge Theorem
(SAS Inequality Theorem)
 If two sides of one triangle are congruent to two sides of
another triangle, and the included angles are not congruent,
then the longer third side is opposite the larger included
angle
Converse of the Hinge
Theorem
 If two sides of one triangle are congruent to two sides of
another triangle, and the third sides are not congruent, then
the larger included angle is opposite the longer third side.
Find the range of possible
values for x
Chapter 7 Similarity
 Ratios and Proportions
 Similar Polygons
 Proving Triangles Similar
 Similarity in Right Triangles
 Proportions in Triangles
Similar Figures
 Have the same shape but not necessarily the same
size
 Is similar to is abbreviated by ~ symbol
 Two Polygons are similar if corresponding angles are
congruent and the corresponding sides are
proportional
Finding Lenghts
Angle Angle Similarity (AA~)
 If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are
similar
Side Angle Side Similarity
(SAS~)
 If an angle of one triangle is congruent to an angle of
a second triangle, and the sides that include the two
angles are proportional then the triangles are similar
Side Side Side Similarity
(SSS~)
 If the corresponding sides of two triangles are
proportional, then the triangles are similar
Are the Triangles Similar? If so
write a similarity statement.
Geometric Mean
 Proportions in which the means are equal
 For numbers a and b, the geometric mean is the
positive number x such that:
 a =x
x b
 Then you cross multiply and solve for x
Theorem – Geometric Mean
 The length of an altitude to the hypotenuse of a right
triangle is the geometric mean of the lengths of the
segments of the hypotenuse.
From the first example
What are the values of x and
y?
What are the values of x and
y?
Side-Splitter Theorem
 If a line is parallel to one side of a triangle and
intersects the other two sides, then it divides those
sides proportionally
Find the value of x
Corollary to the Side Splitter
Thm
 If three parallel lines intersect two transversals, then
the segments intercepted on the transversals are
proportional
Triangle Angle Bisector Thm
 If a ray bisects an angle of a triangle, then it divides
the opposite side into two segments that are
proportional to the other two sides of the triangle
Find the value of x
Chapter 8
Pythagorean Theorem
 In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the
length of the hypotenuse.
45 – 45 – 90 Triangle
 In a 45 – 45 – 90 Triangle, both legs are congruent
and the length of the hypotenuse is √2 times the
length of a leg.
30 – 60 – 90 Triangle
 The length of the hypotenuse is twice the length of
the shorter leg. The length of the longer leg is √3
times the length of the shorter leg.
Trigonometric Ratios
Find the value of w
Using Inverses
 What is the measure of <X to the nearest degree?
Angle of Elevation and
Angle of Depression
 The angle of elevation and the angle of depression are
congruent to each other.
Law of Sines
 Relates the sine of each angle to the length of the
opposite side
 Use when you know AAS, ASA, or SSA
 SSA is generally used for obtuse triangles
Law of Sines
 Relates the sine of each angle to the length of the
opposite side
 Use when you know AAS, ASA, or SSA
 SSA is generally used for obtuse triangles
Law of Cosines
 Relates the cosine of each angle to the side lengths
of the triangle
 Use when you know SAS or SSS
 Find MN to the nearest tenth
Translating Figures
 To translate a figure in the coordinate plane,
translate each point the same units left/right and
up/down.
 For example each point of ABCD is translated 4 units
right and 2 units down. So each (x, y) pair is mapped
to (x+4, y-2)
 Written as:
Properties of Reflections
 Preserve Distance and Angle Measure
 Reflections map each point of the preimage to one
and only one corresponding point of its image
90 Degree Rotation
180 Degree Rotation
270 Degree Rotation
Dilations
Combinations
Find the Area of the
Nonagon
 What is the area of a regular pentagon with 4in
sides? Round your answer to the nearest square in.
 A tabletop has the shape of a regular decagon with a
radius of 9.5 in. What is the area of the tabletop to
the nearest square inch?
Finding Area
 Suppose you want to find the area of a triangle.
What formula could you come up with to find the
area of any triangle using a trig function
 sinA = h/c
 h = c sinA
 A = ½(bc)sinA
What is the area of the
triangle