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Geometry Study Guide
(Play Geometry Jeopardy online at https://jeopardylabs.com/play/geometry-jeopardy436)
Chapter 7
Angles:
An angle is made up of two rays which join at a point called a vertex. The symbol for angle is <.Angles can be named by their vertex or by three letters (<B or <ABC).
Acute angles measure less than 90°.
Right angles measure exactly 90° (the symbol for a right angle is a little box at the vertex).
Obtuse angles measure more than 90° but less than 180°. Reflex angles measure more than 180° but less than 360°. Straight line is exactly 180°.
Vertical & Adjacent Angles:
Two intersecting lines will form 4 angles with a common vertex.
Vertical Angles – Angles that are opposite & equal (congruent)
Adjacent Angles – Angles that are next to each other. They share a common side and vertex.<1 and <3 are vertical (opposite and equal - 60°) <2 and <4 are vertical (opposite and equal - 120°)
<1 and <4 are adjacent
<1 and <2 are adjacent
<2 and <3 are adjacent
<3 and <4 are adjacent
Complementary & Supplementary Angles:
Complementary Angles – two adjacent angles that have a sum of 90°.
Supplementary Angles – two adjacent angles that have a sum of 180° (remember supplementary & straight both start with an s) *Using the above the following are supplementary: <1 and < 2, <2 and <3, <3 and <4, <1 and <4Lines that remain the same distance apart and will never intersect are called parallel lines. The symbol for parallel lines is (‖). Lines that intersect to form right angles are called perpendicular lines. A line that intersects two or more lines is called a transversal (in the example at the right “line t” is the transversal).
Corresponding Angles:
Formed when a transversal intersects parallel lines. Corresponding angles lie on the same side of a
Corresponding angles are equal (congruent)
Triangles:
Classify by Sides:
Equilateral – 3 equal sides/angles Isosceles – 2 equal sides/angles Scalene – 3 sides with different length Classify by Angles:
Acute Triangle – has 3 acute angles Right Triangle – has one right angle Obtuse Triangle – has one obtuse angle Facts:
THE SUM OF THE 3 INSIDE ANGLES IN A TRIANGLE = 180°
Quadrilaterals:
Any 4 sided shape Facts:
Semicircle – ½ circle
Perimeter / Circumference : The distance “Around the Edge/
Outside” of an object. For circles this is called the circumference. Answer with regular units.
Polygons (straight sided objects): add up all the sides Circles: C= x diameter [or C= (2 x radius)] Semicircle = ( x diameter) + diameter 2
Area:
The amount of space enclosed inside of a flat (2d) object. Answer in square units (units2) Triangle: A = ½ (base x height)
Parallelogram (square/rectangle/quadrilateral): A = base x height Circle: A = r2
Semicircle: A = r2 2
http://slideplayer.com/slide/2563366/ (area of composite figures slideshow-GREAT!!)
Example 1
Perimeter = 10 + 3 + 5 + 6 + 8 = 32 m 3
5 5
Example 2 Perimeter
Top Semicircle: C = x diameter = 3.14 x 2 = 6.28 = 3.14 in 2 2 2
Left Semicircle: C = x diameter = 3.14 x 2 = 6.28 = 3.14 in 2 2 2
*Note – Perimeter will not include the bottom of each semicircle Or the top/right of the square because they are inside the
Types of solids:
Polyhedron – a solid with faces that are polygons (close sided figures). Includes prisms and cylinders.
Prism – a solid that has two congruent (equal), polygon shaped bases, and all the other sides/faces are rectangles. (Types: triangular prisms & rectangular prisms).
Cylinder – a solid that has two parallel bases that are
congruent circles. The middle of a cylinder is called a lateral surface (if it were detached/removed it would be in the shape of a rectangle).
Pyramid – a solid with a polygon shaped base and triangular sides. The sides come together to form a point at the top.
Parts of a Solid
:
Base – the polygon shaped bottom
Face – the flat surface
Edge – a line segment that joins to vertices Vertices – a corner
Lateral Surface – The part of a cylinder between the 2 circular faces. If it were detached/removed from a cylinder it would be in the shape of a rectangle.
Surface Area
:
the sum of areas for all the flat surfaces of a solid. The surface area is like covering the
Steps to Finding Surface Area:
1. List the location of each face in a chart. 2. Find the area of each face based on its shape:
Rectangle: A=length x width Parallelogram: A=base x height Triangle: A= ½ (base x height)
Circle: (remember if given the diameter divide by 2 to find the radius)
Lateral Surface: 2πrh (r=radius, h=height)
3. Add all the areas together. 4. Express answer in units2
Rectangular Prism Surface Area Example:
Cylinder Surface Area Example:
Volume:
the amount of space to fill the inside of a 3D solid. Think of opening a solid & filling it withcubes. The answer is in units3. You can learn many different formulas for volume or just remember this:
Volume = B·h
(where B = the area of the base)V = (Base Area) x Height of Solid
Steps to Finding Volume:
1. Determine what shape the base of the solid is.
2. Find the area of the base using the following formulas Rectangle: A=length x width
Parallelogram: A=base x height Triangle: A= ½ (base x height)
Circle: (remember if given the diameter divide by 2 to find the radius)
3. Multiply the base area in step 2 by the height of the solid. (Careful – the solid may be laying on it’s side, the height will be the length of the rectangular face).
4. Express the answer in units3.
Rectangular Prism Volume Example:
Rectangular Base (length x width) = 5 x 3 = 15 Height of Solid = 4
Volume = (Base Area) x Height of Solid = 15 x 4
Total Surface Area = 12+12+50+50+80=204cm2
= 60cm3
Triangular Prism Volume Example:
Triangular Base (Careful-Solid is laying on its side. The base is the shape connecting the rectangular faces)
A= ½ (base x height of triangle) = ½ (8 x 3 ) = 12
Height of Solid (Careful – Solid is laying on its side. The height of the solid is the length of the rectangular face)
h = 10
Volume = (Base Area) x Height of Solid = 12 x 10
= 120 cm3 Cylinder Volume Example:
