Chapter 5 Relationships in Triangles
5.1 – Bisectors of Triangles
 I can distinguish among altitudes, angle bisectors, perpendicular bisectors, medians, midsegments in triangles and use their
properties to solve problems.
 I can construct perpendicular and angle bisectors of triangles.
 I can distinguish among the centroid, orthocenter, incenter, and circumcenter in a triangle and use the properties of each to
solve problems.
 I can prove points on a perpendicular bisector of a line segment are exactly equidistant from the segment’s endpoints.
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Perpendicular Bisector- A segment that goes from either the vertex or a side of a triangle and hits one side of the triangle. The
side that it hits, it cuts in half (bisects) and is perpendicular to.
Perpendicular Bisector Theorem - If a point on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of the segment. Example: If line segment CD is a perpendicular bisector of line segment AB, then AC = BC.
1.
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Find each measure.
a.BC
b. XY
c. PQ
Angle Bisector – A line that goes from the vertex of a triangle to the opposite side and splits the angle of the vertex in half.
Angle Bisector Theorem - If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Example: If ray BF bisects angle DBE, line segment FD is perpendicular to ray BD and line segment FE is perpendicular to ray BE,
then DF = FE.
2.
Find each measure
Point of Concurrency – point where concurrent lines intersect (point where 3 or more lines meet at a common point)
-Perpendicular Bisectors point of concurrency is called the Circumcenter
-Angle Bisectors point of concurrency is called the Incenter.
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Circumcenter Theorem – The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is
equidistant from the vertices of the triangle. PD, PF, and PE are perpendicular bisectors of AB, AC, and BC. So by the circumcenter
theorem – AP = CP = BP.
Incenter – The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the side of the
triangle. If P is the incenter of triangle, then PD = PE = PF.
3.
Find each measure if S is the incenter of triangle MNP.
a.
SU
d. m∠TNS
b. ST
e. m∠RPU
c. SR
f. m∠SPU
4. Write an equation in slope-intercept form for the perpendicular bisector of the segment with the given endpoints. Justify you
answer. C(-4, 5) and D(2, -2)
Homework – Page 327 – 329 (9 – 14, 21 – 30, 32 – 35, 41)
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5.2 – Medians and Altitudes of Triangles
I can distinguish among altitudes, angle bisectors, perpendicular bisectors, medians, midsegments in triangles and use their
properties to solve problems.
 I can construct medians and altitudes of triangles.
I can distinguish among the centroid, orthocenter, incenter, and circumcenter in a triangle and use the properties of each to
solve problems.
 I can prove the medians of a triangle meet at a point called the centroid
 I can explore additional properties of the Centroid theorem.
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Median – a line segment that goes from a vertex of a triangle to the midpoint of the opposite side of the triangle. It cuts the
side of triangle in half when it hits it – it bisects the side it touches of the triangle. Example: line CD is a median of triangle ABC
so line segment CD bisects line segment AB; therefore, AD = BD.
Point of Concurrency – point where concurrent lines intersect (point where 3 or more lines meet at a common point)
-Medians point of concurrency is called a Centroid
Centroid Theorem – The point at which the medians of a triangle intersect. The centroid is 2/3 of the distance from the
vertex to the midpoint of the opposite side. (ex: AP = 2/3AK, BP = 2/3 BL, and CP = 2/3CJ)
(In turn, the lines from the centroid to the midpoint = 1/3 of the whole line)
1 A.
2 A.
1B. In triangle ABC, Q is the centroid and FC = 15, find FQ and QC
2B. In triangle JKL, RP = 3.5 and JP = 9, find PL and PS.
3. An artist is designing a sculpture that balances a triangle on the top of a pole. In the artist’s design on the coordinate plane,
the vertices are located at (1, 4), (3,0), and (3,8). What are the coordinates of the point where the artist should place the pole
under the triangle so that it will balance?
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Altitudes – a line that goes from the vertex of a triangle to the side opposite the vertex. The altitude is perpendicular to the
side it hits – so it forms a right angle together. The altitude does NOT have to hit the side of the triangle in the middle.
Example: line segment BD is an altitude to triangle ABC. When BD hits line segment AD it is perpendicular to the side.
Point of Concurrency – point where concurrent lines intersect (point where 3 or more lines meet at a common point)
-Altitudes point of concurrency is called the Orthocenter.
Orthocenter – the point at which the lines containing the altitudes of a triangle intersect. (ex: P is the orthocenter of triangle
ABC because lines AF, BG, and CD are altitudes and all meet at point P).
***We will do a WORKSHEET HERE TO SHOW EXAMPLES OF PERPENDICULAR BISECTOR, ANGLE BISECTOR, MEDIAN, AND
ALTITUDE
4. The vertices of triangle HIJ are H(1, 2), I (-3, -3) and J(-5, 1). Find the coordinates of the orthocenter of triangle HIJ.
***Page 337 – MAKE SURE TO KNOW THE CHART ON THIS PAGE!!!!!!
HOMEWORK – Page 338 – 339 (5 – 10, 11, 13, 14, 15, 16 – 19, 24, 25, 34, 35)
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1.
5.3 - Inequalities in One Triangle
 I can recognize and apply properties of inequalities to the measures of the angles of a triangle.
I can recognize and apply properties of inequalities to the relationships between the angles and the sides of a triangle.
Use the Exterior Angle Inequality to list all of the angles that satisfy the stated condition.
(a) measures less than ∠14
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(b) measures greater than ∠5
Angle-Side Inequalities - If a one side is longer than another, then the angle opposite that side is greater than the angle
opposite the shorter side. Basically the angle opposite the longest side has the greatest measure, the angle opposite the side
with the second greatest measure is the second largest, and the angle across from the side with the smallest measure is the
smallest. The same is true for sides based on angles measures.
2. List the angles and sides in order from smallest to largest
3. List the sides of triangle ABC in order from shortest to longest
4. Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in
half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the
dimensions shown, which two ends should she tie?
HOMEWORK – Pages 346 – 348 (9-41 odd, 22, 38, 47)
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1.
5.5 – The Triangle Inequality
I can identify that a polygon is a triangle given three side measures (Triangle Inequality Theorem).
Is it possible to form a triangle with the given side lengths? If not explain why not.
a. 6.5, 6.5, 14.5
b. 6.8, 7.2, 5.1
c. 15yd, 16yd, 30yd
d. 2ft, 8ft, 11ft
2A. In triangle PQR, PQ = 7.2 and QR = 5.2. Which measure can not be PR?
(a) 7
(b) 9
(c ) 11
(d) 13
2B. Which of the following could not be a value of the third side of a triangle?
(a) 7
(b) 13
(c ) 10
(d) 22
3.Determine the possible values of x.
4.Find the range of possible measures of x if each set of expressions represents measures of the sides of a triangle.
8, x, 12
5.Determine whether the given coordinates are the vertices of a triangle. Explain. (pg. 365 #38)
X (1, -3), Y(6, 1) and Z(2,2)
HOMEWORK – Page 363-365 (7 – 17ODD, 20, 21, 25, 27, 29, 35, 37, 39, 41)
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5.6 – Inequalities in Two Triangles
I can apply the Hinge Theorem in triangles.
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Hinge Theorem – If 2 sides of a triangle are congruent to two sides of anther triangle, and the included angle of the first is
larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the
second triangle.
(ex: If AB = FG, AC = FH, and angle A is greater than angle F, then BC is greater than GH)
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Converse of the Hinge Theorem – If two sides of a triangle are congruent to 2 sides of anther triangle, and the third side in
the first triangle is longer than the third side in the second triangle, then the included angle in the first triangle is greater than
the included angle in the second triangle. (ex: If JL = PR, KL = QR and PQ is greater than JK, then angle R is greater than
angle L)
1. Compare the given measures:
a. AD and BD
b. m ∠ABD and m∠ BDC
2.
Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the
examining table, and the doctor raises each leg until the patient experiences back pain. Nitan can tolerate the doctor raising
his right leg 35 degrees and his left leg 65 degrees from the table. Which leg can Nitan raise higher above the table?
3.
Find the range of possible values for ‘a’.
4.
HOMEWORK - pages 372 – 376 (10-22, 23, 25, 31-36, 43)