5.1: Midsegments of Triangles
NOTE: Midsegments are also
to the third side in the triangle.
Example: Identify the 3 midsegments in the diagram.
Examples: Identify three pairs of parallel segments in the diagram.
1. AB
2. BC
3. AC
Write an equation to model this theorem based on the figure.
Examples: Points J, K, and L are the midpoints of the sides of ΔXYZ. J is the midpoint of ̅̅̅̅
𝑋𝑌,
̅̅̅̅ and L is the midpoint of 𝑋𝑍
̅̅̅̅. Draw a diagram to model the situation
K is the midpoint of 𝑌𝑍
with midsegments. Then let XY = 14, JK = 8 and KZ = 5.
4. Find LK.
5. Find YK.
6. Find XZ.
7. Find XL.
8. Find JL.
9. Find YJ.
Finding the values of variables
Remember that the midsegment is always half of the parallel side.
Always set up an equation and fill in what you have and then solve.
Algebra Find the value of x.
10.
11.
12.
Examples: X is the midpoint of MN . Y is the midpoint of ON .
13. Find XZ.
14. If XY = 10, find MO.
15. If mM is 64, find mNXY.
AB is a midsegment of ∆GEF. What is the value of x? 2AB = GF
2(2x) = 20
4x = 20
x=5
Exercises
Find the length of the indicated segment.
1. AC
2. TU
3. SU
4. MO
5. GH
6. JK
Algebra In each triangle, AB is a midsegment. Find the value of x.
7.
8.
9.
10.
11.
12.
5.2: Perpendicular and Angle Bisectors
What have we learned to prove that AP = BP
if the segments were drawn in?
What are the ways to prove that a figure is the perpendicular bisector of a segment?
Examples: Use the figure at the right for Exercises 1–3.
1. What is the value of x?
2. Find AB.
Example 4: Find n
3. Find BC.
Examples: Use the figure at the right for Exercises 5–10.
5. How far is M from KL ?
6. How far is M from JK ?
7. How is KM related to JKL?
8. Find the value of x.
9. Find mMKL.
10. Find mJMK and mLMK.
Examples: Find the indicated measures.
11. x, BA, BC
13. x, IK
12. x, EH, EF
14. x, mUWV, mUWT
5.3: Bisectors in Triangles
The perpendicular bisectors of a triangle are concurrent at a point known
as the
and equidistant from
the
.
What point is the circumcenter in the figure?
What is true about PB, PA and PC?
We could circumscribe a circle about the triangle using what point as the center?
Circumcenters can be inside, outside or on the triangle itself!
Acute
Right
Obtuse
What are the coordinates of the circumcenter of a triangle with vertices A(2,7), B(10,7) and
C(10, 3)
The angle bisectors of a triangle are concurrent at a point known as
the
and equidistant from the
What point is the incenter?
What is true about PX, PY and PZ?
We could inscribe a circle inside the triangle using what point?
.
Example: Find the circumcenter of the triangle with the given vertices.
2) P(1, -5)
Q(4, -5)
R(1, -2)
y
x
Examples: Name the point of concurrency of the angle bisectors.
3)
4)
5)
Examples: Find the value of x.
6)
7)
8)
5.4: Medians and Altitudes
The medians of a triangle are concurrent at a point, known as
the
, that is
the
distance from the vertex to the midpoint of the opposite side.
Finding the Length of a Median
1) Write an equation like the one above
2) Fill in the appropriate values and solve.
The altitudes of a triangle are concurrent at a point known as the
The orthocenter can be inside, outside or on the triangle. Draw
a triangle for each and then construct the orthocenter.
Identify the special segments that create each point of concurrency.
.
Examples: In ∆XYZ, A is the centroid.
1. If DZ = 12, find ZA and AD.
and AY.
2. If AB = 6, find BY
Examples: Is MN a median, an altitude, or neither? Explain.
3.
4.
Examples: In Exercises 6–9, name each segment.
6. a median in ∆STU
7. an altitude in ∆STU
8. a median in ∆SBU
9. an altitude in ∆CBU
10. Q is the centroid of ∆JKL. PK = 9x + 21y.
Write expressions to represent PQ and
QK.
5.
5.5: Indirect Proof
1) What are you trying to prove?
2) What is the opposite of what you are trying to prove?
3) What is the first step of this proof then?
Examples: Complete the first step of an indirect proof of the given statement.
4. There are fewer than 11 pencils in the box.
5.
If a number ends in 0, then it is not divisible by 3.
6.
If a number ends in x, then it is a multiple of 5.
7.
mXYZ < 100
8.
∆DEF is a right triangle.
Identifying Contradictions
Indirect proofs should always lead to a contradiction.
Use your definitions to find a contradiction of two statements.
Examples: Identify the two statements that contradict each other.
9. I. MN || GH
II. MN and GH do not intersect.
III. MN and GH are skew.
10. I. ∆CDE is equilateral.
II. mC and mE have the same measure.
III. mC > 60
11. I. ∆JKL is scalene.
II. ∆JKL is obtuse.
III. ∆JKL is isosceles.
Example: Writing an indirect proof. Supply the missing
reasons.
Complete the indirect proof.
12. Given: S  W
T  V
Prove: TS
VW
Assume temporarily that
.
Then by the Converse of the
, S and W
cannot be
that
. This contradicts the given information
. Therefore, TS must be
VW .
5.6: Inequalities in One Triangle
The measure of the exterior angle of a triangle is
than the measure of either
.
Circle the angles that 1 is greater than according to this theorem.
Example 1: Use the figure to explain why 𝑚∠2 > m∠3.
̅̅̅̅ ≅ ̅̅̅̅
In ⊿𝐴𝐶𝐷, 𝐶𝐵
𝐶𝐷, so by the
Theorem, 𝑚∠2 = m∠
. Since
is an exterior
angle of ⊿𝐴𝐵𝐷, 𝑚∠1 > m∠
by the Corollary to the
. By using substitution
𝑚∠
.
Theorem 5-11
If two sides in a triangle are note congruent, then the longer side lieas opposite the
.
For Exercises 2–3, list the angles of each triangle in order from smallest to largest.
2.
3. ∆XYZ, where XY = 25, YZ = 11, and
XZ = 15.
For Exercises 4–5, list the sides of each triangle in order from shortest to longest.
5. ∆MNO, where mM = 56, mN = 108, and
4.
mO = 16
The sum of the lengths of any two sides in a triangle is
the length of the
.
Fill in the blanks.
Hint: You only need to check the sum of the 2 smallest lengths and make certain it is larger
than the 3rd length given.
Examples: Can a triangle have sides with the given lengths? Explain.
6. 10 in., 13 in., 18 in.
7. 6 m, 5 m, 12 m
8. 11 ft, 8 ft, 18 ft
Finding the Length of a Third Side
We are finding a range of values. The 3rd side will be in between 2 numbers.
Ways to Find:
Set up 3 inequalities using x for the 3rd side
OR
Add the 2 numbers and subtract them.
Examples:The lengths of two sides of a triangle are given. Find the range of possible lengths
for the third side.
13. 4, 8 (Set up inequalities)
14. 13, 8
15. 10, 15
5.7: Inequalities in Two Triangles
Examples: Write an inequality relating the given side lengths. If there is not enough
information to reach a conclusion, write no conclusion.
1. AB and CB
2. JL and MO
3. ST and BT
Finding a Range of Values for Angles
Expression must be greater or less than other angle
AND
Expression must be greater than zero.
Examples: Find the range of possible values for each variable.
4.
5.
Examples: Use the diagram at the right for Exercises 6–8. Complete each comparison with <
or >. Then complete the explanation.
6. mACB
mDCE
7. AB
DE
8. BE
CE
Examples: Write an inequality relating the given angle measures.
9. mM and mR
10. mU and mX