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Unit 5: Triangle Properties

Period: 1

Date: 12/16/16

By: America Santana

                          ___

       Find  CC

How To Find the Median in any Triangle

Median Property

Medians are segments that connect a vertex to the midpoint of the opposite side.

Step 1:

Look at what the question is asking you and find those two points in the triangle.

Step 2:

Find the vertex and draw a line to the opposite side midpoint.

Determine the value of x and AB if BQ is a median    of triangle ABCAQ = 5x - 4, QC = 3x + 12,      and AB = 4x - 11.

Example of Finding Median

AQ = QC

5x-4 = 3x+12

5x = 3x+18

2x = 18

x = 9

Step 1:

B

A

Q

C

AB = 4(9) - 11

AB = 36 - 11

AB = 25

Step 2:

How To Find the Altitude in any Triangle

Altitudes are segments that are formed by drawing a segment from one side of the triangle to the opposite vertex. It is the shortest distance from to the vertex to the opposite side.

Altitude Property

Step 1: 

Find the vertex that was given to you.

 

Step 2:

Draw a line all the wat across to the opposite side to make a 90 degree angle.

Example of Finding Alttitude

R

T

SR is an altitude of triangle RST. Solve for if m<SQR = 5x - 3 and m<TRQ = 4x + 3

Q

S

m<SRQ + m<TRQ = 90

5x-3 + 4x+3 = 90

5x+4x-3+3 = 90

9x+0 = 90

9x + 90

x = 10

How To Find an Angle Bisector in any Triangle

Angle Biscetor Property

Angle Bisectors are segments/rays/lines that bisects and angle of the triangle.

Step 1:

Find the angle that is given.

 

Step 2:

Draw a line that starts at the angle and goes through the opposite side.

m<KJL = m<IJL

x+66 = 9x+2

66 = 8x +2

64 = 8x

8 = x

Example of Finding Angle Bisector

JL is an angle bisector of triangle IJK. m<KJL = x+66, m<IJL = 9x+2, and KJ = 2x-5. Determine x and KJ.

KJ = 2(8)-5

KJ = 16-5

KJ = 11

Perpendicular Bisector Property

How to Find a Perpendicular Bisector in any Triangle

Perpendicular bisectors are segments/rays/lines that pass through the triangle and are perpendicular to one side of the triangle.

Step 1: 

Find the midpoint from the segment that you were given.

 

Step 2:

Draw a line from that midpoint straight up until you pass another side.

 

Step 3:

Draw a square at the bottom corner because you just made a 90 degree angle.

XY is a perpendicular bisector of PR. Determine PR of m<PXY = 8x+2, and PX = 3x-7.

Example of Finding Perpendicular Bisector

R

90 = 9x+2

88 = 8x

11 = x

3(11)-7

33-7

26

Y

PR = 26*2

PR = 52

Q

X

P

Triangle Inequality Theorem

32,51,20

Triangle Inequality Theorem: the sum of any two sides of a triangle must be greater than the third.

32 + 51 > 20  *

32 + 20 > 51  *

20 + 51 > 32  *

AB + BC > AC

AB + AC > BC

AC + BC > AB

 

1 + 2 > 3

1 + 3 > 2

2 + 3 > 1

Examples:

9,3,4

9 + 4 > 3  *

9 + 3 > 4  *

3 + 4 > 9  x

Comparing Angles Using Side Measures and Vice Versa

The mid-size side is opposite the middle angles and vice versa.

Segment AC, AB, and BC.

The longest side is opposite the biggest angle and vice versa.

The shortest side is opposite the smallest angles and vice versa.

Order the sides from shortest to longest.

Segment MT, AM, and AT.

Missing angle = 71

SSS and SAS Inequality Theorems 

Converse of Hinge Theorem (SSS Inequality):

Suppose you have two pairs of congruents sides in two different triangles.

Hinge Theorem (SAS Inequality):

Start with two pairs of congruent sides in two different triangles.

Included angles are different measures.

Triangle ABC is congruent to Triangle DEF.

Included angles are congruent.

Segment BC > Segment EF.

Inequalities Between 2 Trinangles

Compare the angles to the non-congruent sides.

D

m<1  >  m<2

AB > CD

A

33

1

34

2

C

B

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