Theorems and Proofs
Transcript of Theorems and Proofs
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 1
4.1 Theorems and Proofs
Answers
1. A postulate is a statement that is assumed to be true. A theorem is a true statement that can/must
be proven to be true.
2. Statements and reasons.
3. It means that the corresponding statement was given to be true or marked in the diagram.
5. Paragraph, two-column, flow diagram
6.
Statements Reasons
β π΅πΆπ΄ β β π·πΆπ΄ Given
β π΅π΄πΆ β β π·π΄πΆ Given
π΄πΆΜ Μ Μ Μ β π΄πΆΜ Μ Μ Μ Reflexive Property
ΞACB β ΞACD ASA β π΄π΅Μ Μ Μ Μ β π΄π·Μ Μ Μ Μ CPCTC (corresponding parts of congruent
triangles must be congruent) 7. β π΅πΆπ΄ β β π·πΆπ΄ and β π΅π΄πΆ β β π·π΄πΆ because it is marked in the diagram. Also, π΄πΆΜ Μ Μ Μ β π΄πΆΜ Μ Μ Μ because by
the reflexive property, any segment is congruent to itself. ΞACB β ΞACD by π΄ππ΄ β because two
pairs of angles and their included sides are congruent. π΄π΅Μ Μ Μ Μ β π΄π·Μ Μ Μ Μ because they are corresponding
parts and corresponding parts of congruent triangles are congruent.
8.
Given
β π΅π΄πΆ
β β π·π΄πΆ
π΄πΆΜ Μ Μ Μ β π΄πΆΜ Μ Μ Μ
ΞACB β ΞACD
β π΅πΆπ΄ β β π·πΆπ΄
π΄π΅Μ Μ Μ Μ β π΄π·Μ Μ Μ Μ
Reflexive Property Given
π¨πΊπ¨ β
CPCTC
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 2
9. β π»πΊπΌ β β π½πΊπΎ because it is given information. Point G is the center of the circle because it is given
information π»πΊΜ Μ Μ Μ , πΊπΌΜ Μ Μ , πΊπ½Μ Μ Μ , πΊπΎΜ Μ Μ Μ are all radii of the circle, because they are segments that connect the
center of the circle with the circle. π»πΊΜ Μ Μ Μ β πΊπΎΜ Μ Μ Μ and πΊπΌΜ Μ Μ β πΊπ½Μ Μ Μ because all radii are congruent. Ξπ»πΊπΌ β
ΞJGK by SASβ because they are triangles with two pairs of corresponding sides congruent and
included angles congruent.
10.
Statements Reasons
β π»πΊπΌ β β π½πΊπΎ Given
Point G is the center of the circle Given
π»πΊΜ Μ Μ Μ , πΊπΌΜ Μ Μ , πΊπ½Μ Μ Μ , πΊπΎΜ Μ Μ Μ are radii Definition of radii
π»πΊΜ Μ Μ Μ β πΊπΎΜ Μ Μ Μ and πΊπΌΜ Μ Μ β πΊπ½Μ Μ Μ all radii are congruent
Ξπ»πΊπΌ β ΞJGK SASβ 11.
Given
π»πΊΜ Μ Μ Μ , πΊπΌΜ Μ Μ , πΊπ½Μ Μ Μ , πΊπΎΜ Μ Μ Μ are
radii
all radii are
congruent
β π»πΊπΌ β β π½πΊπΎ
Point G is the
center of the circle
Ξπ»πΊπΌ β ΞJGK
π»πΊΜ Μ Μ Μ β πΊπΎΜ Μ Μ Μ and
πΊπΌΜ Μ Μ β πΊπ½Μ Μ Μ
Definition of
radii
Given
SASβ
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 3
12.
13. Square ABCD is given. A square has four congruent sides, so π΄π΅Μ Μ Μ Μ β πΆπ΅ Μ Μ Μ Μ Μ and π΄π·Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ . Also,
π·π΅Μ Μ Μ Μ β π·π΅Μ Μ Μ Μ by the reflexive property since any segment is congruent to itself. The triangles have
three pairs of sides congruent so Ξπ΄π΅π· β ΞπΆπ΅π· by πππ β .
14.
Statements Reasons
Square ABCD Given
π΄π΅Μ Μ Μ Μ β πΆπ΅ Μ Μ Μ Μ Μ and π΄π·Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ Definition of a square
π·π΅Μ Μ Μ Μ β π·π΅Μ Μ Μ Μ Reflexive Property
Ξπ΄π΅π· β ΞπΆπ΅π· πππ β
15. Answers vary, but one example is lawyers in a court room.
Definition of a
square
π«π©Μ Μ Μ Μ Μ β π«π©Μ Μ Μ Μ Μ
Square ABCD
π«πππ β π«πππ
π΄π΅Μ Μ Μ Μ β πΆπ΅ Μ Μ Μ Μ Μ
and π΄π·Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
Given
Reflexive
Property
πΊπΊπΊ β
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 4
4.2 Theorems about Lines and Angles
Answers
1. πβ 1 + πβ 3 = 180Β° and πβ 2 + πβ 3 = 180Β° because two angles that form a line are
supplementary. Since both πβ 1 + πβ 2 and πβ 2 + πβ 3 have a sum of 180Β°, by substitution,
πβ 1 + πβ 3 = πβ 2 + πβ 3. Subtract πβ 3 from both sides and the result is πβ 1 = πβ 2. This
means that β 1 β β 2 because if two angles have the same measure then they are congruent.
2.
3.
Statements Reasons
πΆπ· β‘ is the perpendicular bisector to π΄π΅Μ Μ Μ Μ with D on π΄π΅Μ Μ Μ Μ .
Given
β πΆπ·π΅ and β πΆπ·π΄ are right angles Definition of perpendicular
π΄π·Μ Μ Μ Μ β π΅π·Μ Μ Μ Μ Definition of bisector
β πΆπ·π΅ β β πΆπ·π΄ Right angles are congruent
πΆπ·Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ Reflexive property
ΞπΆπ·π΅ β ΞπΆπ·π΄ SASβ
π΄πΆΜ Μ Μ Μ β π΅πΆΜ Μ Μ Μ CPCTC
Given
β 1 β β 5
transitive
property
β 1 β β 3
Two parallel lines are
cut by a transversal
β 3 β β 5
if lines are parallel then
corresponding angles are
congruent
Vertical angles are
congruent
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 5
4.
Statements Reasons
Two parallel lines are cut by a transversal
Given
β 3 β β 5 Alternate interior angles are congruent if lines are parallel
πβ 3 = πβ 5 Two angles that are congruent have the same measure
πβ 5 + πβ 6 = 180Β° Two angles that form a straight line are supplementary
πβ 3 + πβ 6 = 180Β° Substitution
β 3 and β 6 are supplementary Definition of supplementary
5.
6.
Statements Reasons
Two parallel lines are cut by a transversal Given
β 2 β β 6 If lines are parallel then corresponding angles are congruent
β 6 β β 8 Vertical angles are congruent.
β 2 β β 8 Transitive property. 7. If same side interior angles are supplementary, then lines are parallel.
Given
β 2 β β 3
if corresponding angles are
congruent then lines are parallel
β 1 β β 3 β 1 β β 2
π||π
Transitive property of
congruence
Vertical angles are
congruent
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 6
8.
9. If a point is equidistant from the endpoints of a line segment, then the point is on the perpendicular
bisector of the line segment.
Given
πβ 2 + πβ 3 = 180Β° πβ 1 + πβ 2 = 180Β°
Two angles that form a
line are supplementary
πβ 1 + πβ 2 = πβ 2 + πβ 3
if corresponding angles are
congruent then lines are parallel
π||π
Substitution
πβ 1 = πβ 3
Subtraction
β 1 β β 3
Two angles with the same
measure are congruent
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 7
10.
11. Answers vary. Student could draw three angles such that β 1 and β 2 are complementary and β 1
and β 3 are complementary.
12. Answers vary depending on the picture. Possible answer: πβ 1 + πβ 2 = 90Β° and πβ 1 + πβ 3 =
90Β°.
13.
Statements Reasons
πβ 1 + πβ 2 = 90Β° Given
πβ 1 + πβ 3 = 90Β° Given
mβ 1 + mβ 2 = mβ 1 + mβ 3 Substitution
πβ 2 = πβ 3 Subtraction
β 2 β β 3 If two angles have the same measure then they are congruent.
πΆ is equidistant
from A and B
Given
ΞπΆπ·π΅ β ΞπΆπ΅π΄
SSSβ
πβ πΆπ·π΅ + πβ πΆπ·π΄ = 180Β°
Angles that are congruent
and supplementary must be
right angles.
πΆπ·Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
Reflexive
Property
Definition of
midpoint
π΄πΆΜ Μ Μ Μ β πΆπ΅Μ Μ Μ Μ
π· is the midpoint of π΄π΅Μ Μ Μ Μ
β πΆπ·π΅ β β πΆπ·π΄
π΄π·Μ Μ Μ Μ β π·π΅Μ Μ Μ Μ
Given
Definition of
equidistant
CPCTC Two angles that form a straight
line are supplementary
β πΆπ·π΅ πππ β πΆπ·π΄ are right angles
πΆπ·Μ Μ Μ Μ β₯ π΄π΅Μ Μ Μ Μ
If two lines meet at right
angles then they are
perpendicular.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 8
14. Student could draw three angles such that β 1 and β 2 are supplementary and β 1 and β 3 are
supplementary.
Statements Reasons
πβ 1 + πβ 2 = 180Β° Given
πβ 1 + πβ 3 = 180Β° Given
mβ 1 + mβ 2 = mβ 1 + mβ 3 Substitution
πβ 2 = πβ 3 Subtraction
β 2 β β 3 If two angles have the same measure then they are congruent.
15. Show that alternate interior angles are congruent. Show that corresponding angles are congruent.
Show that same side interior angles are supplementary.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 9
4.3 Applications of Line and Angle Theorems
Answers
1. Answers vary.
2. Answers vary. Possible answer: β π΄π΅πΈ and β π·πΈπ΅.
3. Answers vary. Possible answer: β π΄π΅πΈ and β π΅πΈπΊ.
4. Answers vary. Possible answer: β π΄π΅πΆ and β π·πΈπ΅.
5. No. Corresponding angles are congruent if and only if lines are parallel.
6. Yes, because alternate exterior angles are congruent.
7. Not enough information. Vertical angles don't tell you anything about the parallel lines.
8. Yes, because alternate exterior angles are congruent.
9. No, because corresponding angles are not congruent.
10.
Statements Reasons
πΆ is the midpoint of π΅π·Μ Μ Μ Μ Given
π΄π΅Μ Μ Μ Μ β₯ π·πΈΜ Μ Μ Μ Given
β ABC β β EDC Alternate interior angles are congruent if lines are parallel.
π΅πΆΜ Μ Μ Μ β πΆπ·Μ Μ Μ Μ Definition of midpoint
β π΄πΆπ΅ β β πΈπΆπ· Vertical angles are congruent
Ξπ΄π΅πΆ β ΞπΈπ·πΆ ASAβ
11. Add one more line: π΄πΆΜ Μ Μ Μ β πΆπΈΜ Μ Μ Μ by CPCTC.
12. π΅πΆΜ Μ Μ Μ β₯ π·πΈΜ Μ Μ Μ
13. π₯ = 35Β°
14. π₯ β 14.1Β°
15. π΄π΅ = 10 in
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 10
4.4 Theorems about Triangles
Answers
1.
Statements Reasons
Line π·π΅πΈ β‘ constructed parallel to π΄πΆΜ Μ Μ Μ Parallel postulate
β π·π΅π΄ β β π΄, β πΈπ΅πΆ β β πΆ If lines are parallel then alternate interior angles are congruent
πβ π·π΅π΄ = πβ π΄, πβ πΈπ΅πΆ = πβ πΆ Congruent angles have the same measure
πβ π·π΅π΄ + πβ π΄π΅πΆ + πβ πΈπ΅πΆ = 180Β° Angles that form a straight line have measures that sum to 180Β°
πβ π΄ + πβ π΄π΅πΆ + πβ πΆ = 180Β° Substitution
2.
3. Start by constructing π΄π· β‘ , the angle bisector of β π΄, with F the intersection of π΅πΆ Μ Μ Μ Μ Μ πππ π΄π· β‘ . Because
every angle has only one angle bisector, there is only one such line. β π΅π΄πΉ β β πΆπ΄πΉ because angle
bisectors divide angles into two congruent angles. ABΜ Μ Μ Μ β ACΜ Μ Μ Μ because the triangle is isosceles. AFΜ Μ Μ Μ β
AFΜ Μ Μ Μ because any line segment is congruent to itself. Therefore, Ξπ΄π΅πΉ β Ξπ΄πΆπΉ by ππ΄π β because
two pairs of sides and a pair of included angles are congruent. Because β π΅ and β πΆ are
corresponding parts and corresponding parts of congruent triangles are congruent, β π΅ β β πΆ.
Line π·π΅πΈ β‘ constructed parallel to π΄πΆΜ Μ Μ Μ
πβ π·π΅π΄ + πβ π΄π΅πΆ + πβ πΈπ΅πΆ = 180Β°
Parallel Postulate
β π·π΅π΄ β β π΄, β πΈπ΅πΆ β β πΆ
Substitution
If lines are parallel then
alternate interior angles
are congruent
πβ π·π΅π΄ = πβ π΄, πβ πΈπ΅πΆ = πβ πΆ
Congruent angles have the
same measure
πβ π΄ + πβ π΄π΅πΆ + πβ πΆ = 180Β°
Angles that form a straight
line have measures that
sum to 180Β°
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 11
4.
5. Two angles that form a straight line are supplementary, so πβ π·πΆπ΄ + πβ π΄πΆπ΅ = 180Β°. The sum of
the measures of the interior angles of a triangle is 180Β°, so πβ π΅ + πβ π΄πΆπ΅ + πβ π΄ = 180Β°. Since
both sums are 180Β°, by substitution πβ π·πΆπ΄ + πβ π΄πΆπ΅ = πβ π΅ + πβ π΄πΆπ΅ + πβ π΄. Subtract
πβ π΄πΆπ΅ from both sides and the result is πβ π·πΆπ΄ = πβ π΅ + πβ π΄.
6.
Statements Reasons
πβ π·πΆπ΄ + πβ π΄πΆπ΅ = 180Β° Angles that form a line are supplementary
πβ π΅ + πβ π΄πΆπ΅ + πβ π΄ = 180Β° The sum of the measures of the interior angles of a triangle
180Β°.
πβ π·πΆπ΄ + πβ π΄πΆπ΅ = πβ π΅ + πβ π΄πΆπ΅ + πβ π΄ Substitution
πβ π·πΆπ΄ = πβ π΅ + πβ π΄ Subtraction
7. Since Ξπ΄π·π΅ β ΞπΉπ΅π·, π΄π΅Μ Μ Μ Μ β πΉπ·Μ Μ Μ Μ because they are corresponding parts of congruent triangles and so
must be congruent. Therefore, their lengths must be equal so π΄π΅ = πΉπ·. Since π·πΈ = πΈπΉ and π·πΈ +
πΈπΉ = πΉπ·, by substitution, 2π·πΈ = πΉπ·. Again by substitution, 2π·πΈ = π΄π΅. Divide both sides by 2 and
the result is π·πΈ =1
2π΄π΅.
AFΜ Μ Μ Μ β AFΜ Μ Μ Μ
Construct π΄π· β‘ , the angle bisector of β π΄,
with F the intersection of π΅πΆ Μ Μ Μ Μ Μ πππ π΄π· β‘
Reflexive Property
Isosceles Ξπ΄π΅πΆ
SASβ
Given
ABΜ Μ Μ Μ β ACΜ Μ Μ Μ
Definition of isosceles
triangle
Ξπ΄π΅πΉ β Ξπ΄πΆπΉ
An angle has only one
angle bisector
β π΅π΄πΉ β β πΆπ΄πΉ
Definition of angle
bisector
β π΅ β β πΆ
CPCTC
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 12
8. The converse is "if two angles of a triangle are congruent, then the triangle is isosceles." The
converse is true.
9.
Statements Reasons
β π΅ β β πΆ Given
Construct π΄πΉ β‘ , the angle bisector of β π΄, with F the intersection of
π΅πΆ Μ Μ Μ Μ Μ πππ π΄πΉ β‘
An angle has only one angle bisector
β π΅π΄πΉ β β πΆπ΄πΉ Definition of angle bisector
AFΜ Μ Μ Μ β AFΜ Μ Μ Μ Reflexive Property
Ξπ΄π΅πΉ β Ξπ΄πΆπΉ π΄π΄π β π΄π΅Μ Μ Μ Μ β π΄πΆΜ Μ Μ Μ CPCTC
10.
11. Start by constructing π΄πΉ β‘ , the angle bisector of β π΄, with F the intersection of π΅πΆ Μ Μ Μ Μ Μ πππ π΄πΉ β‘ . Because
every angle has only one angle bisector, there is only one such line. β π΅π΄πΉ β β πΆπ΄πΉ because angle
bisectors divide angles into two congruent angles. AFΜ Μ Μ Μ β AFΜ Μ Μ Μ because any line segment is congruent
to itself by the reflexive property. It is assumed that β π΅ β β πΆ. Therefore, Ξπ΄π΅πΉ β Ξπ΄πΆπΉ by
π΄π΄π β because two pairs of angles and a pair of non-included sides are congruent. Because ABΜ Μ Μ Μ and
ACΜ Μ Μ Μ are corresponding parts and corresponding parts of congruent triangles are congruent, ABΜ Μ Μ Μ β
ACΜ Μ Μ Μ .
β π΅π΄πΉ β β πΆπ΄πΉ
Definition of angle
bisector
AFΜ Μ Μ Μ β AFΜ Μ Μ Μ
Construct π΄π· β‘ , the angle bisector of β π΄,
with F the intersection of π΅πΆ Μ Μ Μ Μ Μ πππ π΄π· β‘
Reflexive Property
ABΜ Μ Μ Μ β ACΜ Μ Μ Μ
AASβ
Given
β B β β C
CPCTC
Ξπ΄π΅πΉ β Ξπ΄πΆπΉ
An angle has only one
angle bisector
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 13
12.
Statements Reasons
Ξπ΄π΅πΆ β Ξπ΅π΄π· Given
β π΄π΅πΆ β β π΅π΄π· CPCTC
Ξπ΄πΈπ΅ is isosceles If two angles of a triangle are congruent the triangle is isosceles
13. It is marked that β πΆπ΄π΅ β β πΆπ΅π΄ and β π·π΄π΅ β β π·π΅π΄. Therefore, ΞπΆπ΄π΅ and Ξπ·π΄π΅ are both
isosceles, with πΆπ΄Μ Μ Μ Μ β πΆπ΅Μ Μ Μ Μ and π·π΄Μ Μ Μ Μ β π·π΅Μ Μ Μ Μ . This means both C and D are equidistant from A and B, so
both C and D must be on the perpendicular bisector of π΄π΅Μ Μ Μ Μ . Because two points define a line,
πΆπ· Μ Μ Μ Μ Μ must be the perpendicular bisector of π΄π΅Μ Μ Μ Μ .
14.
Statements Reasons
Ξπ΄π΅πΆ is isosceles with π΄πΆΜ Μ Μ Μ β πΆπ΅Μ Μ Μ Μ Given
π΄πΆ = πΆπ΅ Congruent segments have the same length
E is the midpoint of π΄πΆΜ Μ Μ Μ and D is the midpoint of πΆπ΅Μ Μ Μ Μ Given
π΄πΈ =1
2π΄πΆ and π΅π· =
1
2πΆπ΅ Definition of midpoint
π΅π· =1
2π΄πΆ
Substitution
π΄πΈ = π΅π· Substitution
π΄πΈΜ Μ Μ Μ β π΅π·Μ Μ Μ Μ Definition of congruent
π΄π΅Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ Reflexive property
β πΆπ΄π΅ β β πΆπ΅π΄ Base angles of isosceles triangles are congruent
ΞπΈπ΄π΅ β Ξπ·π΅π΄ ππ΄π β
15. If the triangle is isosceles, you know one pair of sides and one pair of angles is congruent. You also
know π΅π·Μ Μ Μ Μ β π΅π·Μ Μ Μ Μ by the reflexive property, but those three pieces of information are πππ΄, which is
not a criteria for triangle congruence. Because you don't know where point D is along π΄πΆΜ Μ Μ Μ , you can't
prove that the triangles are congruent. For example, without more information, D could be as
shown in the triangle below:
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 14
4.5 Theorems about Concurrence in Triangles
Answers
1.
Statements Reasons
π΄π·Μ Μ Μ Μ is the angle bisector of β π΄ Given
β π΅π΄π· β β πΆπ΄π· Definition of angle bisector
β ABD, β ACD are right angles Give
β ABD β β ACD right angles are congruent
π΄π·Μ Μ Μ Μ β π΄π·Μ Μ Μ Μ Reflexive property
Ξπ΄π΅π· β Ξπ΄πΆπ· π΄π΄π β π΅π·Μ Μ Μ Μ β π·πΆΜ Μ Μ Μ πΆππΆππΆ
2. The angle bisectors meet in a point.
3. Consider Ξπ΄π΅πΆ with angle bisectors π, π and π.
Lines π and π intersect at a point. This point is equidistant from segments π΄π΅Μ Μ Μ Μ and π΄πΆΜ Μ Μ Μ because it is
on line π, the angle bisector of β π΄. This point is also equidistant from segments π΄π΅Μ Μ Μ Μ and π΅πΆΜ Μ Μ Μ because
it is on line π, the angle bisector of β π΅. Therefore, the point of intersection is equidistant from π΄πΆΜ Μ Μ Μ
and π΅πΆΜ Μ Μ Μ , and so must lie on line π, the angle bisector of β πΆ. Line π intersects lines π and π at the
same point, so the three angle bisectors meet at one point.
4. The circle inscribed in the circle has its center at the incenter.
5. Because the incenter is the center of the inscribed circle.
6. The altitudes meet at a point.
7. The orthocenter is at the vertex of the right angle.
8. The orthocenter is inside the triangle.
9. The orthocenter is outside the triangle.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 15
10. The three points are collinear.
11. The centroid always appears to be in between the circumcenter and the orthocenter.
12.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 17
15.
16. The circle passes through the midpoints of the sides of the circle as well as all the midpoints of the
segments connecting the vertices to the orthocenter.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 18
4.6 Applications of Triangle Theorems
Answers
1. πΆπ = 8 πππ π·π = 4.
2. orthocenter, circumcenter, incenter, centroid
3. centroid
4. incenter
5. circumcenter
6. The circumcenter and orthocenter can be outside of the triangle. The median and incenter are
always inside the triangle.
7. All four points of concurrency are in the same place.
8. π₯ = 30
9. π₯ = 20
10. π₯ = 15
11. π₯ = 2.5, π¦ = 10
12. π₯ = 22
13. π₯ = 10
14. π·πΈ = 5, π΅πΆ = 10, π΄πΈ = 8, π΄π· = 8
15. The perimeter is 10 units.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 19
4.7 Theorems about Quadrilaterals
Answers
1. Definition: a quadrilateral with opposite sides parallel. Additional properties: opposite sides
congruent, opposite angles congruent, diagonals bisect each other.
2. Rectangle: a quadrilateral with four right angles. Additional properties: opposite sides parallel,
opposite sides congruent, diagonals congruent, diagonals bisect each other.
3.
Statements Reasons
π βππππ’π π΄π΅πΆπ· Given
π΄π·Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ Definition of a rhombus
π΄π΅πΆπ· is a parallelogram A rhombus is a parallelogram
π·πΈΜ Μ Μ Μ β πΈπ΅Μ Μ Μ Μ Diagonals of a parallelogram bisect each other
π΄πΈΜ Μ Μ Μ β π΄πΈΜ Μ Μ Μ Reflexive property
Ξπ΄πΈπ· β Ξπ΄πΈπ΅ πππ β πβ π΄πΈπ· = πβ π΄πΈπ΅ πΆππΆππΆ
πβ π΄πΈπ· + mβ π΄πΈπ΅ = 180Β° Two angles that form a straight line are supplementary
πβ π΄πΈπ· + πβ π΄πΈπ· = 180Β° Substitution
πβ π΄πΈπ· = 90Β° Algebra
π΄πΆ β₯ π·π΅ Two lines that meet at a right angle are perpendicular
4.
Statements Reasons
π βππππ’π π΄π΅πΆπ· Given
π΄π·Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ β π·πΆΜ Μ Μ Μ β π΅πΆΜ Μ Μ Μ Definition of a rhombus
π΄πΆΜ Μ Μ Μ β π΄πΆΜ Μ Μ Μ πππ π΅π·Μ Μ Μ Μ β π΅π·Μ Μ Μ Μ Reflexive property
Ξπ΄π·πΆ β Ξπ΄π΅πΆ and Ξπ΄π·π΅ β ΞπΆπ·π΅ πππ β
β π·π΄πΈ β β π΅π΄πΈ, β π΄π΅πΈ β β πΆπ΅πΈ, β π΅πΆπΈ β β π·πΆπΈ, β πΆπ·πΈ β β π΄π·πΈ
πΆππΆππΆ
The diagonals bisect the angles Definition of bisect
5. Definition: A quadrilateral with four congruent sides. Additional properties: opposite sides parallel,
diagonals perpendicular, diagonals bisect angles, opposite angles congruent, diagonals bisect each
other.
6. A square is both a rectangle and a rhombus by definition.
7. A square is also a parallelogram since rectangles (and rhombuses) are parallelograms.
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 20
8. A square is a quadrilateral with four right angles and four congruent sides. Additional properties are
diagonals bisect each other and are congruent, diagonals are perpendicular, diagonals bisect angles,
and opposite sides are parallel.
9.
Statements Reasons
πΎππ‘π π΄π΅πΆπ· Given
π΄π·Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ and π·πΆΜ Μ Μ Μ β π΅πΆΜ Μ Μ Μ Definition of a kite
π΄πΆΜ Μ Μ Μ β π΄πΆΜ Μ Μ Μ Reflexive property
Ξπ΄π·πΆ β Ξπ΄π΅πΆ πππ β
10.
Statements Reasons
πΎππ‘π π΄π΅πΆπ· Given
Ξπ΄π·πΆ β Ξπ΄π΅πΆ One diagonal of a kite divides the kite into two congruent triangles
β π· β β π΅ CPCTC
11.
Statements Reasons
πΎππ‘π π΄π΅πΆπ· Given
Ξπ΄π·πΆ β Ξπ΄π΅πΆ One diagonal of a kite divides the kite into two congruent triangles
β π·π΄πΆ β β πΆπ΅π΄ and β π·πΆπ΄ β β π΅πΆπ΄ CPCTC
π΄πΆΜ Μ Μ Μ bisects β π΄ and β πΆ Definition of bisect
12.
Statements Reasons
πΎππ‘π π΄π΅πΆπ· Given
π΄π·Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ Definition of a kite
β π·π΄πΈ β β π΅π΄πΈ One diagonal of a kite bisects its angles
π΄πΈΜ Μ Μ Μ β π΄πΈΜ Μ Μ Μ Reflexive property
Ξπ΄πΈπ· β Ξπ΄πΈπ΅ πππ β π·πΈΜ Μ Μ Μ β π΅πΈΜ Μ Μ Μ CPCTC
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 21
13.
Statements Reasons
πΎππ‘π π΄π΅πΆπ· Given
π΄π·Μ Μ Μ Μ β π΅π΄Μ Μ Μ Μ Definition of a kite
β π·π΄πΈ β β π΅π΄πΈ One diagonal of a kite bisects its angles
π΄πΈΜ Μ Μ Μ β π΄πΈΜ Μ Μ Μ Reflexive property
Ξπ΄πΈπ· β Ξπ΄πΈπ΅ πππ β πβ π΄πΈπ· = πβ π΄πΈπ΅ πΆππΆππΆ
πβ π΄πΈπ· + mβ π΄πΈπ΅ = 180Β° Two angles that form a straight line are supplementary
πβ π΄πΈπ· + πβ π΄πΈπ· = 180Β° Substitution
πβ π΄πΈπ· = 90Β° Algebra
π΄πΆ β₯ π·π΅ Two lines that meet at a right angle are perpendicular
14. A kite is a quadrilateral with two pairs of adjacent congruent sides. Additional properties are one
pair of opposite congruent angles, diagonals perpendicular, one diagonal bisected by another
diagonal, one diagonal bisects its angles.
15.
Statements Reasons
πππππππππππππ π΄π΅πΆπ· Given
π΄πΆΜ Μ Μ Μ β π΅π·Μ Μ Μ Μ Given
π΄π·Μ Μ Μ Μ β π΅πΆΜ Μ Μ Μ Parallelograms have opposite sides congruent
π·πΆΜ Μ Μ Μ β πΆπ·Μ Μ Μ Μ Reflexive property
Ξπ΄πΆπ· β ΞBCD πππ β πβ π΄π·πΆ = πβ π΅πΆπ· πΆππΆππΆ
π΄π·Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ Definition of a parallelogram
πβ π΄π·πΆ + πβ π΅πΆπ· = 180Β° Same side interior angles are supplementary when lines are parallel
πβ π΄π·πΆ + πβ π΄π·πΆ = 180Β° and πβ π΅πΆπ· +πβ π΅πΆπ· = 180Β°
Substitution
πβ π΄π·πΆ = 90Β° and πβ π΅πΆπ· = 90Β° Algebra
πβ πΆπ΅π΄ = 90Β° and πβ π·π΄π΅ = 90Β° Opposites angles of a parallelogram are congruent and therefore have the same measure
β π΄π·πΆ, β π΅πΆπ·, β πΆπ΅π΄, β π·π΄π΅ are right angles
Definition of right angles
ABCD is a rectangle Definition of a rectangle
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 22
16.
Statements Reasons
ππ’πππππππ‘ππππ π΄π΅πΆπ· Given
β π΄ β β πΆ and β π΅ β β π· Given
πβ π΄ = πβ πΆ and πβ π΅ = πβ π· Congruent angles have the same measure
πβ π΄ + πβ π΅ + πβ πΆ + πβ π· = 360Β° Sum of interior angles of a quadrilateral is 360Β°
πβ π΄ + πβ π΅ + πβ π΄ + πβ π΅ = 360Β° and πβ π΄ + πβ π· + πβ π΄ + πβ π· = 360Β°
Substitution
mβ A + mβ B = 180Β° and πβ π΄ + πβ π· =180Β°
Algebra
β π΄ and β π΅ are supplementary; β π΄ and β π· are supplementary
Definition of supplementary
π΄π·Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ and π΄π΅Μ Μ Μ Μ β₯ π·πΆΜ Μ Μ Μ If same side interior angles are supplementary then lines are parallel
π΄π΅πΆπ· is a parallelogram Definition of a parallelogram
Chapter 4 β Reasoning and Proof Answer Key
CK-12 Geometry Honors Concepts 23
4.8 Applications of Quadrilateral Theorems
Answers
1. False
2. True
3. True
4. True
5. Square
6. Parallelogram
7. Rectangle
8. Rectangle
9. Parallelogram; x=4
10. Rhombus, x=120
11. Kite,π₯ = 2β2
12. π₯ = 50Β°
13. In Guided Practice #3, you proved that Ξπ΄πΊπΉ β ΞπΊπ΄π·. This means that β πΉπΊπ΄ β β π·π΄πΊ. Because
alternate interior angles are congruent, πΉπΊΜ Μ Μ Μ β₯ π΄π·Μ Μ Μ Μ .
14. Conjecture: EHGF is a rhombus.
15. Because all angles are right angles, β π΅ β β πΆ β β π· β β π΄. Because it is rectangle, opposite sides
are congruent. Then because E, H, G, F are midpoints, π΅πΈΜ Μ Μ Μ β πΈπΆΜ Μ Μ Μ β π΄πΊΜ Μ Μ Μ β πΊπ·Μ Μ Μ Μ and π΅πΉΜ Μ Μ Μ β πΉπ΄Μ Μ Μ Μ β πΆπ»Μ Μ Μ Μ β
π»π·Μ Μ Μ Μ . This means that ΞπΉπ΅πΈ β Ξπ»πΆπΈ β Ξπ»π·πΊ β ΞπΉπ΄πΊ by SASβ . This means that πΉπΈΜ Μ Μ Μ β πΈπ»Μ Μ Μ Μ β
π»πΊΜ Μ Μ Μ β πΊπΉΜ Μ Μ Μ because they are corresponding parts of congruent triangles. Therefore, EHGF is a
rhombus because it has four congruent sides.