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Congruent Triangles
Triangle Inequality 4.7
Holt Geometry
Congruent Triangles
• Find the range of values for the third side of a triangle,
with 2 sides known.
• Order the lengths of the sides of a triangle, given the
angles
• Order of the angle measures of a triangle, given the
lengths of the sides
homework
Holt Geometry
Congruent Triangles
Finding the range of values for the
length of the third side of a triangle:
Since the third side cannot be larger than the other two
added together, we find the maximum value by adding
the two sides.
Since the third side and the smallest side cannot be
larger than the other side, we find the minimum value
by subtracting the two sides.
Example: Given a triangle with sides of length 3 and 8, find the range of
possible values for the third side.
Difference < Third Side < Sum
8 – 3 < Third Side < 8 + 3
Range of the third side is 5 < x < 11.
Holt Geometry
homework
Congruent Triangles
Inequalities Within a Triangle
If the measures of three sides of a triangle are
unequal, then the measures of the angles
opposite those sides are unequal in the same
order.
P
11
M
8
13
L
LP < PM < ML
mM < mL < mP
Holt Geometry
homework
Congruent Triangles
Inequalities Within a Triangle
If the measures of three angles of a triangle are
unequal, then the measures of the sides opposite
those angles are unequal in the same order.
W
45°
75°
K
60°
J
mW < mJ < mK
JK < KW < WJ
homework
Holt Geometry
Congruent Triangles
Inequalities Within a Triangle
The longest side is
BC
So, the largest angle is
The largest angle is
A
L
So, the longest side is
MN
homework
Holt Geometry
Congruent Triangles
Inequalities Within a Triangle
Can 16, 10, and 5 be the measures
of the sides of a triangle?
Select the third side.
Difference < Third Side < Sum
16 – 10 < Third Side < 16 + 10
6 < Third Side < 26
No! Because 5 is not greater than 6
homework
Holt Geometry
Congruent Triangles
Determine if the three numbers can be
measures of the sides of a triangle.
If no, explain.
a. 13, 28, 19 Yes, 15 < 19 < 41
28 – 13 < Third Side < 13 + 28
b. 9, 4, 4
NO, 5 < 4 < 13
9 – 4 < Third Side < 9 + 4
c. 9, 7, 18
NO, 2 < 18 < 16
9 – 7 < Third Side < 9 + 7
Holt Geometry
homework
Congruent Triangles
If two sides of a triangle have the
following measures, find the range of
possible measures of the third side.
a. 10, 7
10 – 7 < x < 10 + 7
3 < x < 17
b. 18 , 11
18 – 11 < x < 18 + 11
7 < x < 29
homework
Holt Geometry
Congruent Triangles
Write the angles in order from
least to greatest.
A
12
B
52
43
C
C, A, B
homework
Holt Geometry
Congruent Triangles
Write the sides in order from greatest
to least.
Y
128º
22º
30º
Z
X
XZ, XY, YZ
homework
Holt Geometry
Congruent Triangles
Write the angles in order from
least to greatest.
a. S, R, T
b. W, Y, Z
c. S, T, R
homework
Holt Geometry
Congruent Triangles
Is it possible to form a triangle with the given
side lengths? If not, explain why not.
a.
NO
b.
YES
c.
e.
NOd.
NO
f.
NO
YES
Find the range for the measure of the third side
of a triangle given the measures of two sides.
g.
4 < x < 12 h.
6 < x < 16
i.
k.
Holt Geometry
j.
1.5 < x < 6.9
l.
2¾ < x < 3¾
5.4 < x < 13
5¼ < x < 10
homework
Congruent Triangles
Write the sides in order from least to greatest.
58
44
60
a. DE, CE, CD
c. BC, AB, AC
b. BC, AC, AB
109
39
46
32
d. MP, LM, LP
70
44
e. MN, LM, LN
f. QW, MQ, MW
55
41
42
55
g. MN = LM, LN
Holt Geometry
89
49
h. YZ, XZ, XY
49
i. MQ, MP, PQ
homework
Congruent Triangles
Assignment
Geometry:
Inequalities in One Triangle
and
Triangle Inequality
Holt Geometry
Congruent Triangles
Midsegment 4.8
Holt Geometry
Congruent Triangles
• To review midsegments in Trapezoids.
homework
Holt Geometry
Congruent Triangles
If a quadrilateral is a Trapezoid
homework
Holt Geometry
Congruent Triangles
MN is the midsegment of trapezoid PQRS.
What is x? What is MN?
M = ½(b1 + b2)
2x + 11 = ½(8x – 12 + 10)
2x + 11 = ½(8x – 2)
2•6 + 11 = 23
2x + 11
2x + 11 = 4x – 1
N
M
-2x
-2x
8•6 -12 = 36
8x – 12
11
=
2x
–
1
S
P
+1
+1
•How can you check your answer?
12 = 2x
10 + 36 = 46
2
2 homework
= 23
2
2
6=x
Q
Holt Geometry
10
R
Congruent Triangles
Find the length of the midsegment and the
bases of the trapezoid.
3•5+5 = 20
M = ½(b1 + b2)
2x + 16 = ½(3x + 5 + 5x + 7)
2x
+
16
=
½(8x
+
12)
=
26
2•5+16
2x + 16
2x + 16 = 4x + 6
-2x
-2x
5•5+7 = 32
5x + 7
16 = 2x + 6
-6
-6
10 = 2x
homework
2
2
5=x
3x + 5
Holt Geometry
Congruent Triangles
Find x in the trapezoids with the
indicated midsegments.
a. x = 7
b. x = 2
homework
Holt Geometry
Congruent Triangles
Assignment
Geometry:
Midsegments
Holt Geometry