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Math 010: Chapter 9 Geometry Lines, figures, & triangles

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### Math 010: Chapter 9GeometryLines, figures, & triangles

November 25, 2013

9.1 Intro to Geometry (Lines & Angles)

- Lines have infinite length, they go on forever
- Line segments have a finite length
- The length of a segment is denoted by the two endpoints. AB = distance between A and B
- AD = length of the whole line segment

Know how to construct & solve this equation

- If AD = 12 cm, AB = 5 cm, and CD = 4 cm, find the length of BC.
- 5cm x 4cm
- 5 + x + 4 = 12
- x + 9 = 12
- x = 3
- Final Answer: BC = 3 cm

Solve a supplementary angles equation

- 180˚ is a straight line
- Supplementary angles add up to 180˚
- Think straight = supplementary
- What is the value of b?
- 45˚ +39˚ + b + 24˚ = 180˚
- b + 108 = 180
- b = 72˚

Complementary angles equation

- Complementary angles add up to 90˚
- Solve for x.
- (x+3)˚ + (2x – 3)˚ = 90˚
- x˚ +3˚ + 2x˚ – 3˚ = 90˚
- 3x˚ = 90˚
- x = 30˚

Angles: Types of angles

- 1. Acute angles are smaller than 90 degrees
- Examples: 10˚, 45˚, 80˚
- 2. Rightangles are 90 degrees
- Perpendicular lines are lines that form a right angle
- 3. Obtuse angles are larger than 90 degrees and smaller than 180 degrees
- Examples: 100˚, 160˚, 95˚

Vertical angles are congruent

- Congruent angles have equal measure.
- Vertical angles are the angles formed across from each other by two intersecting lines.
- Also note that 134˚ and 46˚ are supplementary

Parallel lines and transversals

- Parallel lines are lines that will never intersect no matter how long you draw them.
- A transversal is a line that intersects two other lines at different points
- Alternate interior angles are shown here:
- AIA’s are congruent!

Know how to fill in all angle measures

- Given: <1 measures 110˚
- Note that <1 and <2 are supplementary
- So <2 measures 70˚
- All angles in this picture measure either 110˚ or 70˚

Triangle equation

- All angles in a triangle add up to 180˚
- Find C.
- 38˚ + 85˚ + C = 180˚
- 123˚ + C = 180˚
- C = 57˚

9.2 Plane Geometric Figures

- Polygons are shapes made up of 3 or more line segments: triangles, rectangles, octagons, etc.
- Circles, ovals are not polygons.
- A regular polygon is a polygon where all sides are equal, and all angles are equal.
- Know this: a pentagon has 5 sides. A hexagon has 6 sides.

hexagon

pentagon

Types of triangles

- Know what an isosceles, equilateral, scalene, and right triangle are.

A right triangle has one right (90˚) angle.

Perimeter

- The perimeteris the distance around the outside of a figure.
- To find the perimeter of a polygon, add up all the side lengths.
- Perimeter of this rectangle

= 2 cm + 6 cm + 2 cm + 6 cm = 16 cm

Circumference

- Circumference is the distance around a circle.
- C = 2πr or πd
- Find the circumference of a circle with diameter 10.
- Circumference = 10π
- Find the circumference of a circle with radius 2.
- Circumference = 2π2 = 4π

Area of a circle

- First need to square r (order of operations)
- Find the area of a circle with radius 5.
- 5 squared is 25
- A = 25π
- Remember the two circle formulas
- Area is the one containing “squared”

9.3 Triangles

- The hypotenuse of a right triangle is the side opposite the right angle.
- Pythagorean Theorem: where c is the hypotenuse.
- Use this theorem with the “3-4-5” triangle
- On exam, show this process to find the value of the hypotenuse.

Similar triangles

- Similar means same shape
- Does not mean same size
- Angle measures same
- Side lengths proportional
- Know how to find missing side
- Multiplication
- We know 14 = 7 · 2;
- 12 = 6· 2
- So, 10 · 2 = 20

Congruent triangles

- Same size and shape – the exact same triangle
- Rules to remember: ASA, SAS, SSS
- Be able to identify which rule applies

SAS

Quiz

- Overall, rate how confident you feel (1-5, 5 best) about the following:
- Geometry vocab
- Lines and angles equations
- Area formulas
- Similar triangles (proportion)
- Congruent triangles rules
- If <1 = 60˚, find the measures of all other angles (2 through 8).

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