Theorem 5-1
Theorem 5-2
Opposite sides of a parallelogram are congruent
Opposite angles of a parallelogram are congruent
A
A
B
ABCD is a parallelogram.
B
ABCD is a parallelogram.
D
C
D
C
1. Label the congruent parts of the parallelogram based
on the theorem.
1. Label the congruent parts of the parallelogram based
on the theorem.
2. If AC = 37 and BD = 5x - 18, find the value of x.
2. ∠A and ∠C are ________________________________ .
3. If ∠B = 9y - 3 and ∠C = 7y + 7, find the value of y.
3. If AB = 7x – 31 and CD = 7x – 24, find the value of x.
4. If ∠B = 3z + 2 and ∠D = 8z + 2, find the value of z.
What does this mean?
Theorem 5-3
Theorem 5-4
Diagonals of a parallelogram bisect each other
If both of the opposite sides of a quadrilateral are
A
B
congruent, then the quadrilateral is a parallelogram
X
ABCD is a parallelogram.
C
D
1. Label the congruent parts of the parallelogram based
on the theorem.
2. What does the word “bisect” mean?
3. If CX = 2a – 1 and BX = 3a – 24, find the value of a.
4. If AD = 50 and AX = 4b + 3, find the value of b.
Use the following diagram to answer the questions.
Z
W
1. State what needs to be true to
prove that WXYZ is a parallelogram
according to Theorem 5-4
Y
X
2. Draw a quadrilateral that has two pairs of congruent
sides but is not a parallelogram.
3. If ZW = 2x-10 and YX = -4x +50, find the value of x.
Theorem 5-5
Theorem 5-6
If one pair of opposite sides of a quadrilateral is both
If both pairs of opposite angles of a quadrilateral are
congruent and parallel, then the quadrilateral is a
congruent, then the quadrilateral is a parallelogram
parallelogram
Use the following diagram to answer the questions.
Use the following diagram to answer the questions.
W
Z
W
Z
1. State what needs to be true to
prove that WXYZ is a parallelogram
according to Theorem 5-4
1. State what needs to be true to
prove that WXYZ is a parallelogram
according to Theorem 5-4
Y
X
X
Y
2. Draw a quadrilateral that has one pair of congruent
sides and one pair of parallel sides but is not a
parallelogram.
2. Draw a quadrilateral that has two pairs of congruent
angles but is not a parallelogram.
3. If ∠WYX = 100° and ∠ZWY = 4y, find the value of y.
3. If ∠Z = 20z + 10 and ∠X = 30z – 10, find the value of z.
Theorem 5-7
Theorem 5-9
If the diagonals of a quadrilateral bisect each
If three parallel lines cut off congruent segments on
other, then the quadrilateral is a parallelogram
one transversal, then they cut off congruent segments
on every transversal
Use the following diagram to answer the questions.
W
Z
1. State what needs to be true to
prove that WXYZ is a parallelogram
according to Theorem 5-4
O
Y
2. Draw a quadrilateral that has two congruent
diagonals.
X
1. If AC = 30, find the value of BC.
2. If XY = 22, find the value of XZ.
3. If XY = 2t – 3 and YZ = 3t – 26, find the value of t.
3. If ZX = 30 and OZ = 2x – 7, find the value of x.
4. If BC = 2g + 6 and AC = 5g – 5, find the value of g.
Theorem 5-10
Theorem 5-11
A line that contains the midpoint of one side of a
The segment that joins the midpoints of two sides of
triangle and is parallel to another side passes through
a triangle
the midpoint of the third side
1.
is parallel to the third side
2. is half as long as the third side
1. If AN = 6, find AC.
1. If BC = 20, find MN.
2. If m∠ANM = 3x – 4 and m∠NCB = 29, find the value of x.
3. If AB = 96 and AM = 3k + 12, find the value of k.
4. If AN = 2j – 8 and NC = j + 11, find the value of j.
2. If m∠AMN = 4i + 4 and m∠MBC = 5i - 3 , find the value of i.
3. If AC = 9d and NC = 5d – 2, find the value of d.
4. What do we know about the angles of ∆𝐴𝑀𝑁 𝑎𝑛𝑑 ∆𝐴𝐵𝐶?
Theorem 5-12
Theorem 5-13
The diagonals of a rectangle are congruent
The diagonals of a rhombus are perpendicular
Theorem 5-16
Theorem 5-14
If an angle of a parallelogram is a right angle,
Each diagonal of a rhombus bisects two angles of the
rhombus
then the parallelogram is a rectangle
J
K
Theorem 5-17
If two consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus
N
Q
M
L
R
T
1. If JL = 2x + 7 and MK = 5x – 35, find the value of x.
P
2. If MN = 2x + 2 and MK = 5x – 5, find JL.
S
1. If m∠QTR = 5v, find the value of v.
2. What type of triangle is ∆𝑄𝑅𝑆 ?
3. Label all congruent segments on the diagram above.
4. List all sets of congruent triangles.
3. What type of triangle is ∆𝑃𝑇𝑆 ?
4. If m∠QPS is 90o, what type of quadrilateral is PQRS?
5. Label all congruent segments and angles of PQRS.
Theorem 5-18
Theorem 5-19
Base angles of an isosceles trapezoid are congruent
The median of a trapezoid
1. is parallel to the bases
G
F
2. has a length equal to the average of the base
lengths
R
S
H
E
2. Angle E and angle F are ___________________________ .
T
E
1. Label all congruent segments and angles in EFGH.
N
L
1. If m∠RET = 5f + 1 and m∠ENL = 3f + 41 , find the value of f.
3. Angle E and angle H are __________________________ .
4. If FE = 36 and GH = 3x – 15, find the value of x.
2. If RS = 47 and NL = 31, what is ET?
5. If m∠FEH = 7n – 3 and m∠GHE = 6n + 6, find the value of n.
3. If RE = 3x + 7 and EN = 5x – 1, find the value of x.
Kite theorem 1
Kite theorem 2
Diagonals of a kite are perpendicular
Exactly one pair of opposite angles is congruent
U
T
X
and exactly one pair of opposite angles is bisected
U
by the diagonal
V
T
X
V
W
1. Identify all pairs of congruent triangles.
W
1. Name the pair of opposite angles that is congruent.
2. Identify all isosceles triangles.
2. Name the diagonal that bisects opposite angles.
3. If m∠UXT = 6c, find the value of c.
3. If m∠UTV = 7q and m∠WTV = 9q – 14, find the value of q.
4. If UX = 3g + 7 and XW = 5g – 4, find UW.
4. If m∠TUV = 9r and m∠TWV = 11r – 22, find the value of r.