Indirect proof
Write indirect proof for the following
1.
2.
Name
LESSON
5-6
Date
Class
Practice C
Inequalities in Two Triangles
A parallelogram is a quadrilateral with two sets of congruent parallel
sides. The opposite angles in a parallelogram are congruent.
Consider parallelogram ABCD.
1. Describe what happens to the diagonals if mA and mC are increased without
changing any side lengths.
_
_
The length of BD increases, and the length of AC decreases.
2. Give the range of lengths for a diagonal in a parallelogram with side lengths of a and b.
between zero and (a b)
Find the range of values for x.
3.
X—
4.
X
X—
—
Z
Y
—
1 (y z)
6 x __
3
10 x 58
Use the figure for Exercises 5 and 6. BC DC.
(Note: The figure is not drawn to scale.)
_
X
"
X—
_
5. Can BD be longer than DC ? If so, find the range
of values for x. If not, explain your answer.
!
_
_
188
Yes, BD can be longer than DC ; 2 x ____
_
$
X—
#
5
_
6. Can DC be longer than BC ? If so, find the range of values for x. If not, explain
your answer.
_
_
No, DC cannot be longer than BC ; possible answer: the inequalities lead
4 and greater than 2.
to the contradiction that x must be both less than __
3
_ _ _ _ _ _
7. Put the segments in order from shortest to longest.
X—
&
AB, CD, DE, FA, EF, BC
_
"
!
Use the figure for Exercises 7 and 8. The intersection
point of the segments is the center of the circle.
(Note: The figure is not drawn to scale.)
8. Name the segment that is congruent to the radius of the circle.
X—
X—
X—
X —
X—
%
#
X —
$
DE
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
45
Holt Geometry
Name
Date
Class
Name
Practice A
LESSON
5-6
LESSON
5-6
Inequalities in Two Triangles
Fill in the blanks to complete the theorems.
�
15
15
10
�
63°
10
�
�
�
47°
�
�
�
5.5
3. AB and DE
�
4.
�
5.
m�UTV � m�WTV
�
two settings of the compass are subject to the Hinge Theorem. To draw
a larger-diameter circle, the measure of the hinge angle must be made
larger. To draw a smaller-diameter circle, the measure of the hinge
130°
110°
Diagram 1
angle must be made smaller.
Diagram 2
Diagram 1
Date
Holt Geometry
Class
Name
5-6
A parallelogram is a quadrilateral with two sets of congruent parallel
sides. The opposite angles in a parallelogram are congruent.
Consider parallelogram ABCD.
_
2. Give the range of lengths for a diagonal in a parallelogram with side lengths of a and b.
PN � QR
�
Use the figure for Exercises 5 and 6. BC � DC.
(Note: The figure is not drawn to scale.)
_
5. Can BD be longer than DC ? If so, find the range
of values for x. If not, explain your answer.
_
188
Yes, BD can be longer than DC ; 2 � x � ____
5
_
_
��
PR � PR
�
�
_
�
m�NPR � m�QRP
�
���
���
�
�
�
�
�
�
�
�
�
�
��
��
���������
�
��
�
�
�
���
�
���
�
�
�
���������
�
�
�
�
��
�
�
�
�
1. TV and XY
2. m�G and m�L
Use the figure for Exercises 7 and 8. The intersection
point of the segments is the center of the circle.
(Note: The figure is not drawn to scale.)
�
����
����������
�
8. Name the segment that is congruent to the radius of the circle.
����������
����������
����������
�� �������
���������
�
�
�
��
�
��
�
AB, CD, DE, FA, EF, BC
��
�
�
�
�
m�G � m�L
TV � XY
No, DC cannot be longer than BC ; possible answer: the inequalities lead
to the contradiction that x must be both less than _4_ and greater than 2.
3
_
�
Compare the given measures.
_
_ _ _ _ _ _
�
If �K is larger than
_�G, then side LM is
longer than side HJ.
So NR � PQ by the Hinge Theorem.
6. Can DC be longer than BC ? If so, find the range of values for x. If not, explain
your answer.
7. Put the segments in order from shortest to longest.
�
Since two sides are congruent and �NPR is smaller
than �QRP, the side across from it is shorter than
the side across from �QRP.
6 � x � _1_ (y � z)
3
10 � x � 58
Example
�
Compare NR and PQ in the figure at right.
���
�
���
Inequalities in Two Triangles
The Converse
_ of the Hinge Theorem is also true. In the example above, if side LM is longer
than side HJ, then you can conclude that �K is larger than �G. You can use both of these
theorems to compare various measures of triangles.
Find the range of values for x.
������
Holt Geometry
_
between zero and (a � b)
�
Class
Theorem
The length of BD increases, and the length of AC decreases.
���������
Date
Reteach
Hinge Theorem
If two sides of one triangle are congruent
to two sides of another triangle and the
included angles are not congruent, then the
included angle that is larger has the longer
third side across from it.
1. Describe what happens to the diagonals if m�A and m�C are increased without
changing any side lengths.
_
44
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
LESSON
4.
x�4
form a triangle, but the lengths of the legs cannot change. Therefore any
��4
Inequalities in Two Triangles
�
������
Possible answer: The legs of a compass and the length spanned by it
60°
22
70°
_
���
������
8. You have used a compass to copy and bisect segments and angles and to draw
arcs and circles. A compass has a drawing leg, a pivot leg, and a hinge at the
angle between the legs. Explain why and how the measure of the angle at the
hinge changes if you draw two circles with different diameters.
Practice C
_
���
�2 � x � 10.5
15
43
2
3� � 6
x�1
12. Warren and his dad are preparing to go sailing for the first
time this year. The two diagrams show the boat’s mast in
different positions as they use a winch to raise it. Notice
that the length of the mast and the distance from the
bottom of the mast to the winch are the same in each
diagram. Tell whether the length of the cable from the
winch to the top of the mast is longer in Diagram 1 or
in Diagram 2.
_
3
7.
x � 15
�x�
���������
17
_5_ � x � ___
37.5
28 � 2x � 2
���������
���
4 � z � 26
3.
(� � 12)°
(3� � 5)°
118°
111°
11. Find the range of values for z in the figure.
5-6
11
54
10. Combine your answers from Exercises 8 and 9 to find the range of values for x.
LESSON
12
7 � x � 58
9. Any angle in a triangle must have a measure greater than 0°.
Solve this inequality for x: 2x � 2 � 0
Name
QR � ST
6.
153°
�
8. Solve your inequality for Exercise 7 for x.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
AB � DE
PS � PQ
6. Compare m�UTV and m�WTV.
1
3. QR and ST
(3� � 21)°
�
7. Rewrite your answer to Exercise 6 by
replacing m�UTV and m�WTV with
the values from the figure.
1
3 –4
�
�
2. AB and DE
�
�
10
�
Find the range of values for x.
6
5. PS and PQ
�
95°
�
40°
3 1–4
43°
45
m�I � m�L
Complete Exercises 6–10 to find the
range of values for x.
�
12
�
�
m�K � m�M
6
4. m�I and m�L
AB � DE
�
�
�
85° 10
�
1. m�K and m�M
120°
100° �
6
�
6
12
5.5
�
14
9
�
�
�
12
9
Inequalities in Two Triangles
2
Compare the given measures.
�
Practice B
2 �
2. If two sides of one triangle are congruent to two sides of another triangle and
included angle
is
the third sides are not congruent, then the larger
across from the longer third side.
�
Class
Compare the given measures.
1. If two sides of one triangle are congruent to two sides of another triangle and the
third side
is
included angles are not congruent, then the longer
across from the larger included angle.
�
Date
�
��
�
�
�
3. AB and AD
4. m�FHE and m�HFG
m�FHE � m�HFG
AB � AD
�� ��������
�
DE
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
45
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
77
46
Holt Geometry
Holt Geometry
Name
T E K S G.8.C
LESSON
5-7
Date
Class
Problem Solving
The Pythagorean Theorem
1. It is recommended that for a height of
20 inches, a wheelchair ramp be
19 feet long. What is the value of
x to the nearest tenth?
2. Find x, the length of
the weight-lifting incline
bench. Round to
the nearest tenth.
XFT
FT
FT
IN
FT
XFT
18.9 ft
4.1 ft
3. A ladder 15 feet from the base of a
building reaches a window that is
35 feet high. What is the length of
the ladder to the nearest foot?
4. In a wide-screen television, the ratio of
width to height is 16 : 9. What are the
width and height of a television that has
a diagonal measure of 42 inches? Round
to the nearest tenth.
width 36.6 in.; height 20.6 in.
38 ft
Choose the best answer.
!USTIN
5. The distance from Austin to San Antonio
is about 74 miles, and the distance from
San Antonio to Victoria is about 102 miles.
Find the approximate distance from Austin
to Victoria.
A 28 mi
B 70 mi
MI
3AN!NTONIO
C 126 mi
D 176 mi
MI
6ICTORIA
6. What is the approximate perimeter of
䉭DEC if rectangle ABCD has a length
of 4.6 centimeters?
F
G
H
J
5.1 cm
6.5 cm
9.8 cm
11.1 cm
!
%
"
$
4.6 cm
#
7. The legs of a right triangle measure 3x
and 15. If the hypotenuse measures
3x 3, what is the value of x?
A 12
B 16
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
8. A cube has edge lengths
of 6 inches. What is the
approximate length of a
diagonal d of the cube?
C 36
D 221
F 6 in.
G 8.4 in.
35
D
H 10.4 in.
J 12 in.
Holt Geometry
Name
T E K S G.8.C
LESSON
5-7
Date
Class
Problem Solving
The Pythagorean Theorem
1. It is recommended that for a height of
20 inches, a wheelchair ramp be
19 feet long. What is the value of
x to the nearest tenth?
2. Find x, the length of
the weight-lifting incline
bench. Round to
the nearest tenth.
XFT
FT
FT
IN
FT
XFT
18.9 ft
4.1 ft
3. A ladder 15 feet from the base of a
building reaches a window that is
35 feet high. What is the length of
the ladder to the nearest foot?
4. In a wide-screen television, the ratio of
width to height is 16 : 9. What are the
width and height of a television that has
a diagonal measure of 42 inches? Round
to the nearest tenth.
width 36.6 in.; height 20.6 in.
38 ft
Choose the best answer.
!USTIN
5. The distance from Austin to San Antonio
is about 74 miles, and the distance from
San Antonio to Victoria is about 102 miles.
Find the approximate distance from Austin
to Victoria.
A 28 mi
B 70 mi
MI
3AN!NTONIO
C 126 mi
D 176 mi
MI
6ICTORIA
6. What is the approximate perimeter of
䉭DEC if rectangle ABCD has a length
of 4.6 centimeters?
F
G
H
J
5.1 cm
6.5 cm
9.8 cm
11.1 cm
!
%
"
$
4.6 cm
#
7. The legs of a right triangle measure 3x
and 15. If the hypotenuse measures
3x 3, what is the value of x?
A 12
B 16
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
8. A cube has edge lengths
of 6 inches. What is the
approximate length of a
diagonal d of the cube?
C 36
D 221
F 6 in.
G 8.4 in.
35
D
H 10.4 in.
J 12 in.
Holt Geometry
Name
Date
LESSON
5-8
Class
Practice C
Applying Special Right Triangles
Multiply and simplify. Assume a and b are nonnegative.
a b)(
a b) 1. (
a b2
a2 b
2. (a b )(a b ) Find the value of x in each figure. Give your answers in simplest radical form.
3.
30°
X
4.
5.
60°
X4
7.
X
2 3 2
4
X
A°
X
X4
6.
X4
2A°
X
83 12
8.
30°
45°
X
X4
60°
4 2 4
X
45°
1
1
2
1
__
2
Greg is a modeling enthusiast. He is working on modeling some
geometric shapes, but he finds he doesn’t have a ruler to take
measurements. In Greg’s desk drawer, he finds a protractor, a
straightedge, and a pencil. For Exercises 9 and 10, use 30°-60°-90°
and/or 45°-45°-90° triangles to accomplish each task.
9. Describe how Greg can draw an exact 2 : 1 replica of a 45°-45°-90° triangle.
That is, he will draw a triangle that has double the length of each side in the
original triangle. (Hint: Look back at Exercise 8.)
Possible answer: Use one of the legs of the original 45°-45°-90° triangle
as the shorter leg of a 30°-60°-90° triangle. The hypotenuse of the
30°-60°-90° triangle will then have twice the length of one of the legs
of the 45°-45°-90° triangle. Then draw a 45°-45°-90° triangle with a leg
as the hypotenuse of the 30°-60°-90° triangle. This larger 45°-45°-90°
triangle has legs with exactly twice the length of the original 45°-45°-90°
triangle.
10. Describe how Greg can draw an exact 1 : 3 replica of a 30°-60°-90°
triangle. Sketch an example.
rj
X nX
Possible answer: Name the length of the longer leg in X
3
___
a 30°-60°-90° triangle x. The shorter leg has length
x.
3
Use the shorter leg of the original triangle as the longer leg of another
30°-60°-90° triangle. The shorter leg of this second triangle then has
1x. Use that leg as the longer leg of a third 30°-60°-90° triangle.
length __
3
This smallest triangle has sides that are exactly one-third the length of
the original.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
61
Holt Geometry
Name
Date
Class
Name
Practice A
LESSON
5-8
5-8
�
1. The sum of the angle measures in a triangle is 180°.
Find the missing angle measure. Then use the
Pythagorean Theorem to find the length
���
of the hypotenuse.
�
�
�
45°; �2
�
3.
�
45°
2
45° �
4.
�
45°
10��
2
4��
2
2� 2
16
�
45°
60°; �3
30°
�
�
4
6. x �
y�
�
4� 3
7. x �
�
10
�
60°
60°
7
14
y�
8. x �
30°
�
�
10� 3
20
y�
9. Andre is building a structure out of playing cards. Each card
is 6.3 centimeters long. He tries leaning the cards against
each other so that the angle at the top is 90°. Find the
distance between the edges of the cards to the nearest tenth.
10. Andre tries leaning the cards against each
other so the angle at the top is 60°. Find the
height x of the tops of the cards.
90°
6.3 cm
90°
�
���
5.5 cm
triangle whose hypotenuse is the length of one of the legs of the larger
�
57�2 inches or about 10 inches, so
triangle. The height of the alcove is _____
8
He can probably not lay a card across the top of the structure in
Exercise 10 because 6.3 cm is the distance between two consecutive
the statues could have been placed in the alcoves.
peaks, and there should be some overlap for the card to stay.
Name
LESSON
5-8
Date
Holt Geometry
Class
Name
LESSON
5-8
Applying Special Right Triangles
Multiply and simplify. Assume a and b are nonnegative.
a�b
2
�
30°
�
4.
5.
60°
Theorem
�
��4
Example
45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, both legs are
congruent and the length of the hypotenuse
�
is � 2 times the length of a leg.
��4
�°
2�°
�
Holt Geometry
Class
Applying Special Right Triangles
2
Find the value of x in each figure. Give your answers in simplest radical form.
3.
Date
Reteach
a �b
�
2. (a � �b)(a � � b) �
60
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Practice C
a � b)(��
a � b) �
1. (��
3
y�
perpendicular to the hypotenuse. This makes another smaller 45°-45°-90°
Andre cannot lay a
card across the top of the structure in Exercise 9 because 6.3 cm � 8.9 cm.
59
�
�3
6. x �
Possible answer: To find the height of a 45°-45°-90° triangle, draw a
11. Tell whether Andre can lay another card across the peaks of
the structures he built in Exercises 9 and 10. Possible answer:
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
8� 3
y�
inches
16
tall. She wonders whether the statues might have been placed in the alcoves. Tell
whether this is possible. Explain your answer.
������
����� ���
��
�
4�3
8. Lucia also finds several statues around the building. The statues measure 9
8.9 cm
���
������
5. x �
Possible answer: Lucia’s hypothesis cannot be correct. The base of the
�
57�2 inches or just over 20 inches long, so a 22 _1_-inch tablet
alcove is _____
4
8
could not fit.
7
___
For Exercises 9 and 10, use a calculator to find each answer.
6.3 cm
�
20� 3
7. Around the perimeter of the building, Lucia finds small alcoves at regular intervals carved
into the stone. The alcoves are triangular in shape with a horizontal base and two sloped
equal-length sides that meet at a right angle. Each of the sloped sides measures 14 _1_
4
inches. Lucia has also found several stone tablets inscribed with characters. The stone
tablets measure 22 _1_ inches long. Lucia hypothesizes that the alcoves once held the stone
8
tablets. Tell whether Lucia’s hypothesis may be correct. Explain your answer.
7��
3
�
y�
2� °
2��
3
Lucia is an archaeologist trekking through the jungle of the Yucatan
Peninsula. She stumbles upon a stone structure covered with creeper
vines and ferns. She immediately begins taking measurements of her
discovery. (Hint: Drawing some figures may help.)
�
8
30
4. x �
�
�
�°
60°
�
In a 30°-60°-90° triangle, the hypotenuse is the length of the shorter
leg multiplied by 2, and the longer leg is the length of the shorter leg
�
multiplied by �3. Find the values of x and y.
30°
�
�
�
30°
���
�
2
12
10��
3
10
�
�
7� 2
____
2
Find the values of x and y. Give your answers in simplest radical form.
4
5. Find the missing angle measure. Then use the
Pythagorean Theorem to find the length of the
hypotenuse.
60°
�
�
�
45°
�
10
45°
45°
2��
2
7
�
45° �
In a 45°-45°-90° triangle, the legs have equal length and the hypotenuse
�
is the length of one of the legs multiplied by �2. Find the value of x.
2
Applying Special Right Triangles
Find the value of x in each figure. Give your answer in simplest
radical form.
2.
3.
1. 8��2
��
�
Class
Practice B
LESSON
Applying Special Right Triangles
2.
Date
���
���
�
�
���
��
��
���
���
�
�����
��4
�
4
6.
�
2�3 � 2
7.
�
�
8�3 � 12
8.
30°
4�2 � 4
1
1
�
�2
���
_1_
�
Use the 45°-45°-90° Triangle Theorem to find the value of x in �EFG.
2
Every isosceles right triangle is a 45°-45°-90° triangle. Triangle
EFG is a 45°-45°-90° triangle with a hypotenuse of length 10.
Greg is a modeling enthusiast. He is working on modeling some
geometric shapes, but he finds he doesn’t have a ruler to take
measurements. In Greg’s desk drawer, he finds a protractor, a
straightedge, and a pencil. For Exercises 9 and 10, use 30°-60°-90°
and/or 45°-45°-90° triangles to accomplish each task.
�
�
�
Rationalize the denominator.
2.
��
�
���
��
���
�
���
�
�
x � 17 �2
���
��
�
� �
�
��
�
3.
x � 22 � 2
4.
�
�
�
�
30°-60°-90° triangle. The shorter leg of this second triangle then has
length _1_x. Use that leg as the longer leg of a third 30°-60°-90° triangle.
3
This smallest triangle has sides that are exactly one-third the length of
the original.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
�
�
Divide both sides by �2.
1.
���
Possible answer: Name the length of the longer leg in �
�
�3 x.
���
a 30°-60°-90° triangle x. The shorter leg has length ___
3
Use the shorter leg of the original triangle as the longer leg of another
61
Hypotenuse is �2 times the length of a leg.
Find the value of x. Give your answers in simplest radical form.
Possible answer: Use one of the legs of the original 45°-45°-90° triangle
as the shorter leg of a 30°-60°-90° triangle. The hypotenuse of the
30°-60°-90° triangle will then have twice the length of one of the legs
of the 45°-45°-90° triangle. Then draw a 45°-45°-90° triangle with a leg
as the hypotenuse of the 30°-60°-90° triangle. This larger 45°-45°-90°
triangle has legs with exactly twice the length of the original 45°-45°-90°
triangle.
10. Describe how Greg can draw an exact 1 : 3 replica of a 30°-60°-90°
triangle. Sketch an example.
��
�
�
10 � x � 2
�
x� 2
10 � ____
___
�
�
�2
�2
�
5� 2 � x
9. Describe how Greg can draw an exact 2 : 1 replica of a 45°-45°-90° triangle.
That is, he will draw a triangle that has double the length of each side in the
original triangle. (Hint: Look back at Exercise 8.)
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
� ���
�
45°
60°
�
���
�
45°
�
��4
In a 45°-45°-90° triangle, if a leg
length is x, then the hypotenuse
�
length is x �2.
�
x � 4 �2
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
81
�����
���
x � 25
62
Holt Geometry
Holt Geometry
Name
T E K S G.5.D
Date
Class
Problem Solving
LESSON
5-8
Applying Special Right Triangles
For Exercises 1–6, give your answers in simplest radical form.
1. In bowling, the pins are arranged in a pattern
based on equilateral triangles. What is the
distance between pins 1 and 5?
12 3 in. or about 20.8 in.
2. To secure an outdoor canopy, a 64-inch cord is extended
from the top of a vertical pole to the ground. If the cord
makes a 60° angle with the ground, how tall is the pole?
IN
32 3 in. or about 55.4 in.
_
Find the length of AB in each quilt pattern.
4.
3.
"
!
3 in.
30°
! 3 in.
4 in. "
8 3 in. or about 4.6 in.
_____
3 2 in. or about 4.2 in.
3
Choose the best answer.
5. An equilateral triangle has an altitude of
21 inches. What is the side length of
the triangle?
6. A shelf is an isosceles right triangle, and
the longest side is 38 centimeters. What
is the length of each of the other two sides?
14 3 in.
19 2 cm
Use the figure for Exercises 7 and 8.
Assume 䉭JKL is in the first quadrant, with m⬔K 90.
Y
* (2, 7)
_
7. Suppose that JK is a leg of 䉭JKL, a 45-45-90
triangle. What are possible coordinates of point L?
A (6, 4.5)
C (6, 2)
B (7, 2)
D (8, 7)
_
3
+ (2, 2)
X
0
3
8. Suppose 䉭JKL is a 30-60-90 triangle and JK
is the side opposite the 60° angle. What are the
approximate coordinates of point L?
F (4.9, 2)
G (4.5, 2)
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
H (8.7, 2)
J (7.1, 2)
36
Holt Geometry
Name
T E K S G.5.D
Date
Class
Problem Solving
LESSON
5-8
Applying Special Right Triangles
For Exercises 1–6, give your answers in simplest radical form.
1. In bowling, the pins are arranged in a pattern
based on equilateral triangles. What is the
distance between pins 1 and 5?
12 3 in. or about 20.8 in.
2. To secure an outdoor canopy, a 64-inch cord is extended
from the top of a vertical pole to the ground. If the cord
makes a 60° angle with the ground, how tall is the pole?
IN
32 3 in. or about 55.4 in.
_
Find the length of AB in each quilt pattern.
4.
3.
"
!
3 in.
30°
! 3 in.
4 in. "
8 3 in. or about 4.6 in.
_____
3 2 in. or about 4.2 in.
3
Choose the best answer.
5. An equilateral triangle has an altitude of
21 inches. What is the side length of
the triangle?
6. A shelf is an isosceles right triangle, and
the longest side is 38 centimeters. What
is the length of each of the other two sides?
14 3 in.
19 2 cm
Use the figure for Exercises 7 and 8.
Assume 䉭JKL is in the first quadrant, with m⬔K 90.
Y
* (2, 7)
_
7. Suppose that JK is a leg of 䉭JKL, a 45-45-90
triangle. What are possible coordinates of point L?
A (6, 4.5)
C (6, 2)
B (7, 2)
D (8, 7)
_
3
+ (2, 2)
X
0
3
8. Suppose 䉭JKL is a 30-60-90 triangle and JK
is the side opposite the 60° angle. What are the
approximate coordinates of point L?
F (4.9, 2)
G (4.5, 2)
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
H (8.7, 2)
J (7.1, 2)
36
Holt Geometry