WARM UP 2/25/14

Triangle Inequality and Triangle mid-Segment

Angle-Side Relationship

You can list the angles and sides of a triangle from

smallest to largest (or vice versa)

› The smallest side is opposite the smallest angle

› The longest side is opposite the largest angle

Angle-Side Relationship

List the angles of ΔABC in order from smallest to largest.

Answer: C, A, B

Angle-Side Relationship

A. AB. BC. CD. D

List the sides of ΔRST in order from shortest to longest.

A. RS, RT, ST

B. RT, RS, ST

C. ST, RS, RT

D. RS, ST, RT

Angle-Side Relationship

Compare the measures AD and BD.

Answer: mACD > mBCD, so AD > DB.

In ΔACD and ΔBCD, AC BC, CD CD, and ACD > BCD.

Inequalities in Triangles

Compare the measures ABD and BDC.

Answer: ABD > BDC.

In ΔABD and ΔBCD, AB CD, BD BD, and AD > BC.

Inequalities in Triangles

A. AB. BC. CD. D

A. mJKM > mKML

B. mJKM < mKML

C. mJKM = mKML

D. not enough information

B. Compare JKM and KML.

Inequalities in Triangles

Inequalities in One Triangle

6

3 2

6

3 3

4 3

6

Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third.

They have to be able to reach!!

Triangle Inequality Theorem

AB + BC > AC

A

B

C

AB + AC > BC

AC + BC > AB

Example: Determine if the following lengths are legs of triangles

A) 4, 9, 5

4 + 5 ? 9

9 > 9

We choose the smallest two of the three sides and add them together. Comparing the sum to the third side:

B) 9, 5, 5

Since the sum is not greater than the third side, this is not a

triangle

5 + 5 ? 9

10 > 9Since the sum is greater than the third side, this is

a triangle

Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third

side?

B

A

C

6 12

X = ?

12 + 6 = 18

12 – 6 = 6Therefore: 6 < X < 18

Vocabulary

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 midpointsDE is the midsegment

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

DE║AC

1DE AC

2

D

B

C

E

A

Midsegment Theorem

Example 1In the diagram, ST and TU are midsegments

of triangle PQR. Find PR and TU.

PR = ________ TU = ________16 ft 5 ft

Example 2In the diagram, XZ and ZY are

midsegments of triangle LMN. Find MN and ZY.

MN = ________ ZY = ________53 cm

14 cm

Example 3In the diagram, ED and DF are midsegments

of triangle ABC. Find DF and AB.

DF = ________ AB = ________26 52

3X 4

5X+2

x = ________10