Reteach
Perpendicular and Angle Bisectors
Theorem
Example
Each point on is
equidistant from
points F and G.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a
segment, then it is equidistant, or the same
distance, from the endpoints of the segment.
Given: is the perpendicular bisector of FG.
Conclusion: AF  AG
The Converse of the Perpendicular Bisector Theorem is also true. If a point is equidistant
from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
You can write an equation for the perpendicular bisector of a segment. Consider the
segment with endpoints Q(5, 6) and R(1, 2).
Step 2 Find the slope of the  bisector of QR.
Step 1 Find the midpoint of QR.
 x1  x2 y1  y 2   5  1 6  2 

,
 2 ,
2   2
2 

y 2  y1
26
Slope of QR

x2  x1 1   5 

 (2, 4)
2
3
So the slope of the  bisector of QR is
3
.
2
Step 3 Use the point-slope form to write an equation.
y  y1  m(x  x1)
y 4
3
 x  2
2
Point-slope form
Slope 
3
; line passes through (2, 4), the midpoint of QR.
2
Find each measure.
1. RT  ________________
2. AB  ________________
3. HJ  ________________
Write an equation in point-slope form for the perpendicular bisector
of the segment with the given endpoints.
4. A(6, 3), B(0, 5)
________________________________________
5. W(2, 7), X(4, 3)
________________________________________
Name _______________________________________ Date __________________ Class __________________
Reteach
Perpendicular and Angle Bisectors continued
Theorem
Example
Point P is equidistant from
Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is equidistant from the sides of
the angle.
sides
and
Given: MP is the angle bisector of LMN.
Conclusion: LP  NP
LMP  NMP
Converse of the Angle
Bisector Theorem
If a point in the interior of an angle is
equidistant from the sides of the angle,
then it is on the bisector of the angle.
Given: LP  NP
Conclusion: MP is the angle bisector of LMN.
Find each measure.
6. EH
7. mQRS
_____________________
____________________
Use the figure for Exercises 9–11.
9. Given that JL bisects HJK and LK  11.4, find LH.
________________________________________
10. Given that LH  26, LK  26, and mHJK  1228,
find mLJK.
________________________________________
11. Given that LH  LK, mHJL  (3y  19)8, and
mLJK  (4y  5)8, find the value of y.
________________________________________
8. mWXZ
____________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Bisectors of Triangles
Perpendicular bisectors
The point of intersection of
and
are
concurrent because they
intersect at one point.
and
is called
the circumcenter of NPQ.
Theorem
Example
Circumcenter Theorem
The circumcenter of a triangle is
equidistant from the vertices of
the triangle.
Given: MR, MS, and MT are
the perpendicular bisectors
of NPQ.
Conclusion: MN  MP  MQ
If a triangle on a coordinate plane has two sides that lie along the axes, you can easily find the
circumcenter. Find the equations for the perpendicular bisectors of those two sides. The
intersection of their graphs is the circumcenter.
HD, JD , and KD are the perpendicular bisectors of EFG.
Find each length.
1. DG
________________________
3. FJ
________________________
2. EK
________________________
4. DE
________________________
Find the circumcenter of each triangle.
5.
6.
________________________________________
________________________________________
Name _______________________________________ Date __________________ Class __________________
Reteach
Bisectors of Triangles continued
The point of intersection of
and
is called
the incenter of GHJ.
Angle bisectors of GHJ
intersect at one point.
Theorem
Example
Incenter Theorem
The incenter of a triangle is
equidistant from the sides of
the triangle.
Given: AG, AH, and AJ are
the angle bisectors
of GHJ.
Conclusion: AB  AC  AD
WM and WP are angle bisectors of MNP, and WK  21.
Find mWPN and the distance from W to MN and NP.
mNMP  2mNMW
Def. of  bisector
mNMP  2(328)  648
Substitute.
mNMP  mN  mNPM  1808
n Sum Thm.
648  728  mNPM  1808
Substitute.
mNPM  448
Subtract 1368 from each side.
mWPN 
1
mNPM
2
mWPN 
1
(448)  228 Substitute.
2
Def. of  bisector
The distance from W to MN and NP is 21 by the Incenter Theorem.
PC and PD are angle bisectors of CDE. Find each measure.
7. the distance from P to CE
________________________
8. mPDE
_____________________
KX and KZ are angle bisectors of XYZ. Find each measure.
9. the distance from K to YZ
________________________
10. mKZY
_____________________
Name _______________________________________ Date ___________________ Class __________________
Medians and Altitudes of Triangles
and
are medians
of a triangle. They each join a
vertex and the midpoint of the
opposite side.
The point of intersection of
the medians is called the
centroid of nABC.
Theorem
Example
Centroid Theorem
The centroid of a triangle is
2
located
of the distance from
3
each vertex to the midpoint of
the opposite side.
Given: AH, CG, and BJ are medians of nABC.
2
2
2
Conclusion: AN  AH, CN  CG, BN  BJ
3
3
3
In ABC above, suppose AH  18 and BN  10. You can use the Centroid Theorem
to find AN and BJ.
AN 
2
AH
3
Centroid Thm.
BN 
2
BJ
3
Centroid Thm.
AN 
2
(18)
3
Substitute 18 for AH.
10 
2
BJ
3
Substitute 10 for BN.
Simplify.
15  BJ
AN  12
In QRS, RX  48 and QW  30. Find each length.
1. RW
________________________
3. QZ
________________________
2. WX
________________________
4. WZ
________________________
In HJK, HD21 and BK18. Find each length.
5. HB
________________________
7. CK
________________________
6. BD
________________________
8. CB
________________________
Simplify.
Name _______________________________________ Date __________________ Class __________________
Reteach
Medians and Altitudes of Triangles continued
and
are altitudes
of a triangle. They are
perpendicular segments that join
a vertex and the line containing
the side opposite the vertex.
The point of intersection of
the altitudes is called the
orthocenter of nJKL.
Find the orthocenter ofABC with vertices A(–3, 3), B(3, 7), and C(3, 0).
Step 1 Graph the triangle.
Step 2 Find equations of the lines containing two altitudes.
The altitude from A to BC is the horizontal line y  3.
The slope of AC 
03
1
  , so the slope of the altitude
3   3 
2
from B to AC is 2. The altitude must pass through B(3, 7).
y  y1  m(x  x1)
Point-slope form
y  7  2(x  3)
Substitute 2 for m and the coordinates of B(3, 7) for (x1, y1).
y  2x  1
Simplify.
Step 3 Solving the system of equations y  3 and y  2x  1, you find that the coordinates of
the orthocenter are (1, 3).
Triangle FGH has coordinates F(3, 1), G(2, 6), and H(4, 1).
9. Find an equation of the line containing the
altitude from G to FH.
________________________________________
10. Find an equation of the line containing the
altitude from H to FG.
________________________________________
11. Solve the system of equations from Exercises 9
and 10 to find the coordinates of the orthocenter.
________________________________________
Find the orthocenter of the triangle with the given vertices.
12. N(1, 0), P(1, 8), Q(5, 0)
13. R(1, 4), S(5, 2), T(1, 6)
Reteach
The Triangle Midsegment Theorem
Name _______________________________________ Date __________________ Class __________________
A midsegment of a triangle joins the midpoints of two sides of the triangle.
Every triangle has three midsegments.
R is the midpoint of
S is the midpoint of
is a midsegment
of nCDE.
Use the figure for Exercises 1–4. AB is a midsegment ofRST.
1. What is the slope of midsegment AB and the slope
of side ST ?
________________________________________
2. What can you conclude about AB and ST ?
________________________________________
3. Find AB and ST.
________________________________________
4. Compare the lengths of AB and ST .
________________________________________
UseMNP for Exercises 5–7.
5. UV is a midsegment ofMNP. Find the
coordinates of U and V.
________________________________________
6. Show that UV i MN.
________________________________________
________________________________________
________________________________________
7. Show that UV 
1
MN.
2
________________________________________________________________________________________
Name _______________________________________ Date __________________ Class __________________
Reteach
The Triangle Midsegment Theorem continued
Theorem
Example
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to
a side of the triangle, and its length is
half the length of that side.
Given: PQ is a midsegment of nLMN.
1
Conclusion: PQ i LN, PQ  LN
2
You can use the Triangle Midsegment Theorem to
find various measures inABC.
HJ 
1
AC
2
n Midsegment Thm.
HJ 
1
(12)
2
Substitute 12 for AC.
HJ  6
JK 
4
Simplify.
1
AB n Midsegment Thm.
2
HJ i AC
Midsegment Thm.
1
AB Substitute 4 for JK.
2
mBCA  mBJH
Corr. ? Thm.
mBCA  358
Substitute 358 for mBJH.
8  AB
Simplify.
Find each measure.
8. VX  ____________________
9. HJ  ____________________
10. mVXJ  ____________________
11. XJ  ____________________
Find each measure.
12. ST  ____________________
13. DE  ____________________
14. mDES  ____________________
15. mRCD  ____________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Properties of Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides.
All parallelograms, such as
FGHJ, have the following properties.
Properties of Parallelograms
FG  HJ
GH  JF
Opposite sides are congruent.
mF  mG  180°
mG  mH  180°
mH  mJ  180°
mJ  mF  180°
Consecutive angles are supplementary.
F  H
G  J
Opposite angles are congruent.
FP  HP
GP  JP
The diagonals bisect each other.
Find each measure.
1. AB
2. mD
________________________________________
Find each measure in
________________________________________
LMNP.
3. ML
________________________
5. mLPM
________________________
7. mMLN
________________________
4. LP
________________________
6. LN
________________________
8. QN
________________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Properties of Parallelograms continued
You can use properties of parallelograms to find measures.
WXYZ is a parallelogram. Find mX.
mW  mX  180
(7x  15)  4x  180
11x  15  180
11x  165
x  15
If a quadrilateral is a
,
then cons. ? are supp.
Substitute the given values.
Combine like terms.
Subtract 15 from both sides.
Divide both sides by 11.
mX  (4x)  [4(15)]  60
If you know the coordinates of three vertices of a parallelogram, you can use slope
to find the coordinates of the fourth vertex.
Three vertices of
RSTV are R(3, 1), S(1, 5), and T(3, 6).
Find the coordinates of V.
Since opposite sides must be parallel, the rise and the run from
S to R must be the same as the rise and the run from T to V.
From S to R, you go down 4 units and right 4 units. So, from
T to V, go down 4 units and right 4 units. Vertex V is at V(7, 2).
You can use the slope formula to verify that ST RV .
CDEF is a parallelogram. Find each measure.
9. CD
________________________
11. mF
________________________
10. EF
________________________
12. mE
________________________
The coordinates of three vertices of a parallelogram are given.
Find the coordinates of the fourth vertex.
13.
ABCD with A(0, 6), B(5, 8), C(5, 5)
________________________________________
14.
KLMN with K(4, 7), L(3, 6), M(5, 3)
________________________________________
Name _______________________________________ Date ___________________ Class __________________
Conditions for Parallelograms
You can use the following conditions to
determine whether a quadrilateral such
as PQRS is a parallelogram.
Conditions for Parallelograms
QR  SP
QR SP
QR  SP
If one pair of opposite sides is
and , then PQRS is a parallelogram.
P  R
Q  S
If both pairs of opposite angles are ,
then PQRS is a parallelogram.
PQ  RS
If both pairs of opposite sides are , then
PQRS is a parallelogram.
PT  RT
QT  ST
If the diagonals bisect each other, then
PQRS is a parallelogram.
A quadrilateral is also a parallelogram if one of the angles is
supplementary to both of its consecutive angles.
65  115  180, so A is supplementary to B and D.
Therefore, ABCD is a parallelogram.
Show that each quadrilateral is a parallelogram for the given values.
Explain.
1. Given: x  9 and y  4
2. Given: w  3 and z  31
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Conditions for Parallelograms continued
You can show that a quadrilateral is a parallelogram by using any of the conditions
listed below.
Conditions for Parallelograms
• Both pairs of opposite sides are parallel (definition).
• One pair of opposite sides is parallel and congruent.
• Both pairs of opposite sides are congruent.
• Both pairs of opposite angles are congruent.
• The diagonals bisect each other.
• One angle is supplementary to both its consecutive angles.
EFGH must be a parallelogram
because both pairs of opposite
sides are congruent.
JKLM may not be a parallelogram
because none of the sets of conditions
for a parallelogram is met.
Determine whether each quadrilateral must be a parallelogram.
Justify your answer.
3.
4.
________________________________________
________________________________________
________________________________________
________________________________________
6.
5.
________________________________________
________________________________________
________________________________________
________________________________________
Show that the quadrilateral with the given vertices is a parallelogram by
using the given definition or theorem.
7. J(2, 2), K(3, 3), L(1, 5), M(2, 0)
Both pairs of opposite sides are parallel.
8. N(5, 1), P(2, 7), Q(6, 9), R(9, 3)
Both pairs of opposite sides are
congruent.
Name _______________________________________ Date ___________________ Class __________________
Properties of Special Parallelograms
A rectangle is a quadrilateral with four right angles. A rectangle has the
following properties.
Properties of Rectangles
If a quadrilateral is a rectangle, then it
is a parallelogram.
If a parallelogram is a rectangle, then
its diagonals are congruent.
Since a rectangle is a parallelogram, a rectangle also has all the properties of parallelograms.
A rhombus is a quadrilateral with four congruent sides. A rhombus has the following
properties.
Properties of Rhombuses
If a quadrilateral is a
rhombus, then it is a
parallelogram.
If a parallelogram is a
rhombus, then its diagonals
are perpendicular.
If a parallelogram is a
rhombus, then each
diagonal bisects a pair
of opposite angles.
Since a rhombus is a parallelogram, a rhombus also has all the properties of parallelograms.
ABCD is a rectangle. Find each length.
1. BD
________________________
3. AC
________________________
2. CD
________________________
4. AE
________________________
KLMN is a rhombus. Find each measure.
5. KL
________________________
6. mMNK
________________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Properties of Special Parallelograms continued
A square is a quadrilateral with four right angles and four congruent sides.
A square is a parallelogram, a rectangle, and a rhombus.
Show that the diagonals of square HJKL are congruent
perpendicular bisectors of each other.
Step 1
Show that HK  JL .
HK 
JL 
 6  0
 4  2
2
2
  4  2  2 10
2
  0  6   2 10
2
HK  JL  2 10, so HK  JL .
Step 2
Show that HK  JL .
42 1
slope of HK 

60 3
slope of JL 
06
 3
42
Since the product of the slopes is 1, HK  JL .
Step 3
Show that HK and JL bisect each other by comparing their midpoints.
midpoint of HK  (3, 3)
midpoint of JL  (3, 3)
Since they have the same midpoint, HK and JL bisect each other.
The vertices of square ABCD are A(1, 0), B(4, 5), C(1, 8), and D(4, 3).
Show that each of the following is true.
7. The diagonals are congruent.
________________________________________________________________________________________
8. The diagonals are perpendicular bisectors of each other.
________________________________________________________________________________________
Name _______________________________________ Date ___________________ Class __________________
Conditions for Special Parallelograms
You can use the following conditions to determine whether a parallelogram
is a rectangle.
JL  KM
If one angle is a right angle, then
is a rectangle.
JKLM
If the diagonals are congruent, then
is a rectangle.
JKLM
You can use the following conditions to determine whether a parallelogram is a rhombus.
If one pair of consecutive
sides are congruent, then
TUVW is a rhombus.
If the diagonals are
perpendicular, then
TUVW is a rhombus.
If one diagonal bisects a pair of
opposite angles, then
TUVW is
a rhombus.
Determine whether the conclusion is valid. If not, tell what additional
information is needed to make it valid.
1. EFGH is a rectangle.
________________________________________
2. MPQR is a rhombus.
________________________________________
For Exercises 3 and 4, use the figure to determine whether the conclusion
is valid. If not, tell what additional information is needed to make it valid.
3. Given: EF GH, HE FG, EG  FH
Conclusion: EFGH is a rectangle.
________________________________________
________________________________________
4. Given: mEFG  90
Conclusion: EFGH is a rectangle.
________________________________________
________________________________________
Name _______________________________________ Date ___________________ Class __________________
Reteach
Conditions for Special Parallelograms continued
You can identify special parallelograms in the coordinate plane by examining
their diagonals.
If the Diagonals are . . .
. . . the Parallelogram is a
congruent
rectangle
perpendicular
rhombus
congruent and perpendicular
square
Use the diagonals to determine whether
parallelogram ABCD is a rectangle, rhombus,
or square. Give all the names that apply.
Step 1
Find AC and BD to determine whether
ABCD is a rectangle.
AC 
 6  1
BD 
 6  1
Since
Step 2
2
  5  1  41
2
 1  5   41
2
2
41  41, the diagonals are congruent. So ABCD is a rectangle.
Find the slopes of AC and BD to determine whether ABCD is a rhombus.
slope of AC 
5 1 4

6 1 5
slope of BD 
1 5
4

6 1
5
 4  4 
Since       1 , the diagonals are not perpendicular. So ABCD is
 5  5 
not a rhombus and cannot be a square.
Use the diagonals to determine whether a parallelogram with the
given vertices is a rectangle, rhombus, or square. Give all the
names that apply.
5. V(3, 0), W(6, 4), X(11, 4), Y(8, 0)
________________________________________
7. H(1, 3), J(10, 6), K(12, 0), L(3, 3)
________________________________________
6. L(1, 2), M(3, 5), N(6, 3), P(4, 0)
________________________________________
8. E(4, 3), F(1, 2), G(2, 1), H(5, 0)
________________________________________
Name _______________________________________ Date ___________________ Class __________________