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WARM UP 2/25/14
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Triangle Inequality and Triangle mid-Segment
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Angle-Side Relationship
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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
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List the angles of ΔABC in order from smallest to largest.
Answer: C, A, B
Angle-Side Relationship
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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
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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
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Compare the measures ABD and BDC.
Answer: ABD > BDC.
In ΔABD and ΔBCD, AB CD, BD BD, and AD > BC.
Inequalities in Triangles
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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
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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!!
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Triangle Inequality Theorem
AB + BC > AC
A
B
C
AB + AC > BC
AC + BC > AB
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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
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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
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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
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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
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Example 1In the diagram, ST and TU are midsegments
of triangle PQR. Find PR and TU.
PR = ________ TU = ________16 ft 5 ft
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Example 2In the diagram, XZ and ZY are
midsegments of triangle LMN. Find MN and ZY.
MN = ________ ZY = ________53 cm
14 cm
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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