1

Triangle Angle Sum Theorem• The sum of the measures of the angles of a

triangle is 180°. m∠A + m∠B + m∠C = 180

A

B

C

Ex: If m∠A = 30 and m∠B = 70; what is m∠C ?

m∠A + m∠B + m∠C = 180 30 + 70 + m∠C = 180 100 + m∠C = 180 m∠C = 180 – 100 = 80

Exterior Angle Theorem

1

2 3 4

P

Q RIn the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR.interior

Angle 4 is called an _______ angle of ΔPQR.exterior

An exterior angle of a triangle is an angle that forms a _________, (they add up to 180) with one of the angles of the triangle.linear pair

____________________ of a triangle are the two angles that do not forma linear pair with the exterior angle.

Remote interior angles

In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

In ΔPQR, 4 is an exterior angle because 3 + 4 = 180 .

The measure of an exterior angle of a triangle is equal to sum

of its ___________________remote interior angles

Exterior Angle Theorem

1

2

3 4 5

In the figure, which angle is the exterior angle? 5

which angles are the remote the interior angles? 2 and 3

If 2 = 20 and 3 = 65 , find 5

65

20

If 5 = 90 and 3 = 60 , find 2

85

90 60

30

Triangle Inequality Theorem

TriangleInequalityTheorem

The sum of the measures of any two sides of a triangle is

_______ than the measure of the third side.greater

a

b

c

a + b > c

a + c > b

b + c > a

Triangle Inequality Theorem

Can 16, 10, and 5 be the measures of the sides of a triangle?

No! 16 + 10 > 5

16 + 5 > 10

However, 10 + 5 > 16

Medians, Altitudes, Angle Bisectors

Perpendicular Bisectors

A

B

C

Given ABC, identify the opposite side

1. of A.

2. of B.

3. of C.

BC

AC

AB

Just to make sure we are clear about what an opposite side is…..

A new term…

Point of concurrency

• Where 3 or more lines intersect

B

A

C

M

N

L

Definition of a Median of a Triangle

A median of a triangle

is a segment

whose endpoints

are a vertex and a

midpoint of the opposite

side.

The point where all 3 medians intersectThe point where all 3 medians intersect

CentroidCentroidIs the point of Is the point of concurrencyconcurrency

The The centroidcentroid is 2/3 the distance is 2/3 the distance from the vertex to the side.from the vertex to the side.

2x2x

xx

1010

55

3232

XX1616

angle bisector of a triangle

a segment that bisects an angle of the triangle and goes to the opposite side.

The Incenter is where all The Incenter is where all 3 Angle bisectors intersect3 Angle bisectors intersect

Incenter Incenter Is the point of concurencyIs the point of concurency

Any point on an angle bisector is Any point on an angle bisector is equidistance from both sides of the angle equidistance from both sides of the angle

Any triangle has three altitudes.

Definition of an Altitude of a Triangle

A altitude of a triangle is a segment that has one endpoint at a vertex and the other creates a right angle at the opposite side.

The altitude is perpendicular to the opposite side while going through the vertex

Acute Triangle

Orthocenter is where all the Orthocenter is where all the altitudes intersect.altitudes intersect.

OrthocenterOrthocenter

A Perpendicular bisector of a side does A Perpendicular bisector of a side does not have to start at a vertex. It will formnot have to start at a vertex. It will form

a a 90° angles90° angles and bisectand bisect the side. the side.

CircumcenterCircumcenterIs the point of concurrencyIs the point of concurrency

Any point on the Any point on the perpendicular bisectorperpendicular bisectorof a segment is equidistant from theof a segment is equidistant from the

endpoints of the segment.endpoints of the segment.

AA

BB

CC DD

AB is the perpendicularAB is the perpendicularbisector of CDbisector of CD

The Midsegment of a Triangle is a segment that connects the midpoints of

two sides of the triangle.

D

B

C

E

A

D and E are midpoints

DE is the midsegment

The midsegment of a triangle is parallel to the third side and is half as long as that side.

DE AC1

DE AC2

Example 1In the diagram, ST and TU are midsegments of

triangle PQR. Find PR and TU.

PR = ________ TU = ________16 ft 5 ft

Give the best name for ABGive the best name for ABAA

BB

AA

BB

AA

BB

AA

BB

AA

BB||||

|| ||

||||

MedianMedian AltitudeAltitude NoneNone AngleAngleBisectorBisector

PerpendicularPerpendicularBisectorBisector