Post on 15-Mar-2018
Chapter 5Relationships within Triangles
5.1 Midsegment Theorem and Coordinate Proof
5.2 Use Perpendicular Bisectors
5.3 Use Angle Bisectors of Triangles
5.4 Use Medians and Altitudes
5.5 Use Inequalities in a Triangle5.6 Inequalities in Two Triangles and Indirect Proof
Name _________________________________ Block _______
SOL G.5
The student, given information concerning the lengths of sides and/or measures of
angles in triangles, will:
a) order the sides by length, given the angle measures;
b) order the angles by degree measure, given the side lengths;
c) determine whether a triangle exists; and
d) determine the range in which the length of the third side must lie.
These concepts will be considered in the context of real-world situations.
TriangleMidsegment
TriangleMidsegment
A midsegment of a triangle is a segment
connecting the _____________________ of
two sides of the triangles.
Example: ______________
If a segment joins the midpoints of two sides of a triangle, then the
segment is _________________________ to the third side and
_______________ as long.
Using the diagram above, if BD is a midsegment of ∆ACE, then:
1) _______________________ 2) __________________________
Practice: D is the midpoint of AB and E is the midpoint of BC.
A
D
B
E
C
1) If DE = 8, find AC.
2) If AC = 17, find DE.
3) If BC = 9, find EC.
4) If DB = 5, find AB.
5) If m∠BDE = 35˚, find m∠BAC.
6) If m∠BCA = 110˚, find m∠DEB.
Practice: DE is a midsegment of ∆ABC. Find the value of x.
1. 2. 3.
5.1 Midsegment Theorem and Coordinate Proof
Use ∆GHJ, where D, E and F are midpoints of the sides.
1. If DE = 4x + 5 and GJ = 3x + 25, find DE.
2. If EF = 2x + 7 and GH = 5x – 1, what is EF?
You try!
Use ∆ABC, the midpoints are L, M, and N.
LM ∥ ____________ AB ∥ ____________
If AC = 14, then LN = ______________
If MN = 8, then AB = ______________
If NC = 3, then LM = _______________
If LN = 5, then ___________ = 10
If LM = 3x + 1 and BC = 10x – 6, then LM = ___________________
If NM = x – 1 and AB = 3x – 7, then AB = __________________
5.2 Use Perpendicular Bisectors
Perpendicular Bisector
Perpendicular Bisector
A line segment that is perpendicular (creates right angles) to a
side of a triangle and passes through its _______________________.
EquidistantTwo points are the ________________________________ from a third
point, creating congruent segments.
If a point is on the perpendicular bisector of a segment, then it is
___________________________ from the endpoints of the segment.
If ________ is the bisector of ________
then ___________________.
Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a
segment, then the point lies on the
____________________________________.
If __________________ then
_______________________________.
Example: KM is the perpendicular bisector of JL. Find JK and ML.
Example: DC is the perpendicular bisector of AB and AE ≌ BE.
What segment lengths are equal?
Is E on DC? Explain.
Concurrency
Concurrency of Perpendicular Bisectors of a
Triangle
The perpendicular bisectors of a triangle intersect at a point
that is ______________________________ from the vertices of
the triangle.
If PD, PE, and PF are perpendicular
bisectors, then ____________________.
A point of concurrency is the point where three or more lines,
rays, or segments intersect.
The point of intersection of the lines, rays, or segments is
called the ____________________________________________.
In the diagram, the perpendicular bisectors of ∆ABC meet at point G. Find the indicated
measure.
1. Find BG. 2. Find GA.
You try!
1. CD is the perpendicular bisector of AB.
A. What is the relationship between AD and DB?
B. What is the relationship between ∠ADC and ∠BDC?
C. What is the relationship between AC and CB? Explain.
2. In ∆DEG, FH is a perpendicular bisector of DG with H on DG. If DH = 2y + 3, GH = 7y – 42,
and m∠FHG = (x2 + 9)˚, find the value of x and y.
What is the measure of DG?
For questions 3 – 4, circle the correct word or phrase to complete each sentence.
3. Three or more lines that intersect at a common point are called _______________________
lines.
A. Parallel B. Perpendicular C. Concurrent
4. Any point on the perpendicular bisector of a segment is ______________________________
the endpoints of the segment.
A. Parallel to B. Equidistant to C. Congruent to
D
F
H G
E
5.3 Use Angle Bisectors of a Triangle
Recall, an angle bisector is a ray that divides an angle into two congruent adjacent angles.
Angle Bisector
Angle Bisector Theorem
If a point is on the bisector of an angle, then it
is ____________________from the two sides of
the angle.
Since AD is an bisector, then _________________.
If a point is in the interior of an angle and is equidistant from the
sides of the angle, then it lies on the bisector of the angle.
Can you conclude that BD bisects ∠ABC? Explain.
Find the value of x.
4. 5.
1. 2. 3.
Concurrency of Angle Bisectors of a Triangle:
The angle bisectors of a triangle intersect at a
point that is equidistant from the sides of the
triangle.
You try!
Find the value of x.
1. 2. 3.
4. Write an equation of the perpendicular bisector of the segment with endpoints P(-2, 3) and
Q(4, 1). Show all work.
Midpoint of PQ _____________
SlopePQ __________
Slope⊥ _______________
Y-Intercept ⊥ ________________
Equation of Perpendicular Bisector ____________________
30˚
30˚
10
x + 3
(6x + 14)˚
(9x – 1)˚
3x 4x – 9
5.4 Medians and Altitudes
MedianA median of a triangle is a segment from a _______________ to the
_____________________ of the opposite side.
Each triangle has three medians! The medians of a triangle are
concurrent, and the point of concurrency is called the _________________
inside the triangle. Medians do NOT have to be perpendicular!
Concurrency of Medians
The medians of a triangle intersect at a point that is two thirds of the
distance from each vertex to the midpoint of the opposite side.
If P is the centroid of ∆ABC, then AP = AD, BP = BF, CP = CE 23
23
23
Example: P is the centroid of ∆DEF, EH ⊥ DF, DH = 9, DG = 7.5, EP = 8, and DE = FE.
A. Find the length of FH.
B. Find the length of EH.
C. Find the length of PH.
D. Find the perimeter of ∆DEF.
E
J
F
G
DH
P
8
7.5
9
Example: Point G is the centroid of ∆ABC. Use the given information to find the value of x.
FG = x + 8 and GA = 6x – 4
Altitude
**Altitude is often called height
An altitude is a perpendicular segment that connects the
____________________________ to its __________________________.
Altitudes are NOT bisectors! And, they do not have to be inside the ∆.
Concurrency of Altitudes
The lines containing the altitudes of
a triangle are concurrent.
In ∆ABC, is BD a perpendicular bisector, median or altitude?
1. 2. 3.
4. Use the picture to name the following parts of the triangle.
A. Median
B. Angle Bisector
C. Perpendicular Bisector
D. Altitude
You try!
1. Use the picture to name the following parts of the triangle.
A. Median
B. Angle Bisector
C. Perpendicular Bisector
D. Altitude
2. ZC is an altitude, CYW = 9x + 38, and WZC = 4x.
Solve for x.
Find the measures of ∠CZW and ∠CYW.
.5. XW is an angle bisector. ∠YXZ = 20y + 8, ∠WXY = 8y + 12, ∠XZY = 12y. Label the diagram.
A. Solve for y.
B. Find the measure of ∠ZWX.
C. Is XW an altitude? Justify your answer.
Is it a Triangle?
Triangle Inequalities:
• Allow us to order sides or angles of triangles by length
• Do not give us exact lengths; just relative lengths (order)
• Help us to determine whether triangles can be formed
Determine if the following side
lengths could form a triangle. Prove your
answer with an inequality.
Hint: the sum of the two smaller numbers
must be larger than the third number.
1. 8, 17, 24
3. 3, 3, 7
5. 28, 50, 22
2. 24, 12, 11
4. 35, 41, 7
5. 6, 18, 14
Finding a third side
rangeLet x = third side
Given two sides of a triangle, you can set
up an inequality using the sum and difference to show range of possible
lengths for the third
side.
6. 14 and 22
8. 3 and 11
7. 31 and 28
9. 19 and 45
5.5 Use Inequalities in a Triangle
Example: If a triangle has lengths of 15 feet and 27 feet, check all possible lengths for
the third side.
34 ft 12 ft 29 ft 18ft 43 ft 15 ft 52 ft
Ordering Angles
The angles of a triangle can be put in order by comparing the sides.
The smallest angle is always opposite the _____________________________.
The largest angle is always opposite the _____________________________.
Ordering Sides
The sides of a triangle can be put in order by comparing the angles.
The shortest side is always opposite the _____________________________.
The longest side is always opposite the _____________________________.
Largest angle
Largest side
Smallest angle
Smallest side
List the sides and the angles in order from smallest to largest.
1. 2.10
79
RS
T
33˚A B
C
You try!
1. Sketch and label the triangle described.
Side lengths: about 3 m, 7 m, and 9 m, with longest side on the bottom
Angle measures: 16˚, 41˚, and 123˚, with the smallest angle on the left.
2. Is it possible to construct a triangle with the given side lengths? If not, explain why not.
A. 6, 7, 11 B. 3, 6, 9 C. 28, 34, 39
3. Describe, using an inequality, the 4. Describe the lengths of the third side
possible values of x. of the triangle given the lengths of the
other two sides.
A. A. 3, 4 B. 12, 18
5. Find the value of x and use the indicated angle measures to list the sides of ABC in order
from shortest to longest.
m∠A = (9x + 29)˚, m ∠ B = (93 – 5x)˚, and m ∠ C = (10x + 2)˚
x + 112x + 10
5x – 9
5.6 Inequalities in Two Triangles and Indirect Proof
The hinge theorem states that if two sides of one triangle are
congruent to two sides of another triangle, and the included angle
of the first is larger than the included angle of the second, then the
third side of the first triangle is _________________ than the third side
of the second triangle
Hinge Theorem
Hinge Theorem
If two sides of one triangle are congruent to two sides of another
triangle, and the third side of the first is longer than the third side of
the second, then the included angle of the first is __________________
than the included angle of the second.
WX ______ ST
∠C ______ ∠F
Complete each statement with <, > or =.
1. 2. 3.
Write an inequality to describe the restrictions on the value of x.
4. 5.
A
C B
D F
E
105˚
110˚
AB ______ DE
6 6
21
m∠1 ______ m∠2
1
2
8 7
m∠1 ______ m∠2
6
6
x
7
65˚
70˚
12
12
3x + 2
12x – 7
Steps for writing an indirect proof
1.
2.
3.
Identify the statement you want to prove. Assume temporarily that this statement is ____________ by assuming that its opposite is true.
Reason logically until you reach a __________________________.
Point out that the desired conclusion must be ___________ because the contradiction proves the temporary assumption
false.
In other words, you are proving something indirectly by showing that is cannotbe false.
Example 1: Suppose you want to write indirect proof of this statement: “If x is an odd number, then x is not divisible by 4.” What temporary assumption should start your proof?
Example 2: Suppose you want to write indirect proof of this statement: “If x + y ≠ 14, and y = 5, then x ≠ 9.”What temporary assumption should start your proof?
You try!
Complete each statement with <, > or =.
1. 2. 3.
Write an inequality to describe the restrictions on the value of x.
4. 5.
6. Match the conclusion on the right with the given information on the left.
1. AB = BC, m∠1 > m∠2 a. m∠7 > m∠8
2. AE > EC, AD = CD b. AD > AB
3. m∠9 < m∠10, BE = ED c. m∠3 + m∠4 = m∠5 + m∠6
4. AB = BC, AD = CD d. AE > EC
FG ______ LM m∠1 ______ m∠2 m∠1 ______ m∠2
3
3
x + 3
3x + 2
110˚ 27˚
4x – 3
2xA
D
B C