Category Archives: Geometry

Kinds Of Triangles

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In addition to math help previously available here, this year you will find Algebra videos and Geometry videos.   Go ahead.  Watch the Preview here!  If you’d like to have the full video for reference, just download it at Mindbites.com.  You can save it to your computer and access it whenever you need to.  For those of my friends who are totally disinterested in math, you can watch it at 2:00 AM and it is guaranteed to cure your insomnia!

Today’s video is entitled Kinds Of Triangles.  It’s a 12-minute video that shows each type of triangle used in Geometry and takes a look at their angles and sides.  

Triangles are one of the most familar shapes in the world. As simple as a triangle is, there are several different kinds of triangles and they have different properties. We will also look at the interior, the exterior, as well as the triangle itself.

In this video you will learn all about each type of triangle. You will learn about their angles and their sides and the relationships between sides and angles. Make sure you commit all definitions to memory so when you need to write a proof involving a triangle, you will have the tools you need.

Enjoy this lesson. It’s very straightforward and not difficult at all. Just get all the details and terminology! Success is in the details!

How Many Lines of Symmetry? (LOS)

1. Circle has infinite # LOS.
2. Parallelogram has NO LOS.
3. Rhombus has 2 LOS.
4. Reg Hexagon has 6 LOS.
5. Square has 4 LOS.
6. Rectangle has 2 LOS.
7. Reg Pentagon has 5 LOS.
8. Equilateral Triangle has 3 LOS.
9. Kite has 1 LOS.
10. Isosceles Trapezoid has 1 LOS. Non-Isosceles Trap has NO LOS.

Every regular polygon has the same number of LOS as it does sides!

Exterior Angles Of A Triangle

What do you know about the exterior angles of a triangle?

1. Each exterior angle forms a linear pair with its adjacent interior angle.
2. In other words, each exterior angle + adjacent interior angle = 180 degrees
3. Exterior angle = Sum of the two remote interior angles.
4. In other words, an exterior angle = sum of the non-adjacent interior angles.
5. In any regular convex polygon, exterior angles always add to 360 degrees.
6. In any regular convex polygon, each exterior angle = 360/number of sides.
7. You can extend a side of a polygon to form an exterior angle.
8. What else?

30-60-90/45-45-90 Triangle Relationships

30-60-90 Triangles

The side opposite the 30 degree angle is the short side. Let’s start there. This side is half of the hypotenuse…or you could say the hypotenuse is two times the short side. The hypotenuse is the longest side. That leaves the side opposite the 60 degree angle….the middle side. It is always √3 times the short side. In other words:

Short Side = x
Medium Side = √3 x
Long Side = 2x
_____________________________________________________________________
45-45-90 Triangles

The two sides opposite the 45 degree angles are congruent. The third side is √2 times the shorter sides.

Both Congruent Sides = x
Hypotenuse = x √2

Remember Your Right Triangle Trig Ratios?

SOHCAHTOA

Sin A = Opposite/Hypotenuse

Cos A = Adjacent /Hypotenuse

Tan A = Opposite/Adjacent

Remember you can use these ratios if you have a right triangle. Angle A refers to either of the two acute angles in the triangle but not the right angle. If you do not have enough info to use the Pythagorean Theorem, try using these trig ratios to solve the triangle.

What Are Skew Lines? Aren’t You Dying To Know?

Everybody needs to understand skew lines 🙂 …..before they take any end of course test.
Skew lines are lines that do not lie in the same plane. That is, they are noncoplanar. Think of two lines that are not parallel, yet they do not intersect. Those lines are skew lines.

The lines in this drawing are skew lines. They are not in the same plane. The do not intersect. They are not parallel.

Deductive or Inductive?

Way back in the beginning of the year you learned about 2 kinds of reasoning. Later you learned to write proofs. Remember how much fun that was!!!!

Inductive Reasoning: This kind of reasoning is based on patterns that you observe. Here’s an ex.
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
Based on this pattern, what is the sum of the first 9 consecutive positive odd numbers?
a. 49
b. 64
c. 81
d. 100

Answer: c Do you see the pattern?.

Deductive Reasoning: This kind of reasoning is based on facts…theorems, postulates, properties. When you write a proof using theorems, you are using deductive reasoning.

Random Facts On The Geometry EOCT

1. Concentric circles look like a bulls eye. They have the same center.
2. If a figure is enlarged it is called a dilation. The scale factor is greater than 1.
3. If the scale factor is less than 1, the figure gets smaller.
4. Volume is easy. Just find the area of the base, then multiply that by the height. If it is a pyramid or cone, do the same thing, then divide it by 3. Use cubic units.
5. The sum of the exterior angles of a convex polygon is always 360.
6. Geometry uses DEDUCTIVE reasoning and INDUCTIVE reasoning. Deductive reasoning is based on facts, like in proofs. Inductive reasoning is based on patterns.
7. If you have 3 skirts, 4 shirts, and 5 hats, you can make 3x4x5 or 60 different outfits!
8. Lines that are not parallel and are not in the same plane are called SKEW lines.
9. The scale factor is the “ratio of the sides” in the order stated.
10. Parallel lines have the same slope.
11. Any two sides of a triangle must add up to be greater than the third side.
12. A hexagon contains 6 equilateral triangles.
13. The MEAN is the average. The MEDIAN # is the middle number. The MODE is the one that occurs the most frequently. The RANGE is the highest # – the lowest #.

EOCT “Must Knows” In A Nutshell

1. Central angle = intercepted arc
2. Inscribed angle = 1/2 intercepted arc
3. If two sides of a triangle are 3 and 7, the third side must be greater than 4 and less than 10.
4. The midsegment of a triangle connects the midponts of any two sides of a triangle.
5. The midsegment is parallel to the 3rd side.
6. The midsegment = 1/2 of the 3rd side.
7. The midsegment forms 2 triangles that are similar.
8. In an equilateral triangle, the median, altitude, angle bisector, and perp bisector are all the same segment.
9. Triangles contain 180 degrees. Quads contain 360 degrees. Pentagons contain 540 degrees. Hexagons contain 720 degrees. Circles contain 360 degrees.
10. Number of degrees in convex polygon is (n-2)180.
11. Number of degrees in exterior angles of convex polygon is 360.
11. Acute triangle—each angle is less than 90.
12. Supplementary angles add to 180. Complementary angles add to 90.
13. Triangles are congruent by SAS, SSS, ASA, AAS, HL, and LL.
14. Triangles are similar by SSS~, AA~, SAS~
15. Similar triangles means corresponding sides are proportional and corr angles are congruent.
16. Similar triangles have a scale factor.
17. Scale factor is side/side. Use corresponding sides.

Problems #20-24

20. A building that is 50 feet tall casts a shadow 30 feet long. Nearby, a tree casts a 75 foot long shadow. How tall is the tree? (Hint: Use similar triangle ratios) Read it twice. Wording is tricky.

a. 95 ft
b. 110 ft
c. 125 ft
d. 140 ft

Answers: 50/30 =x/75 so x = 125

21. Which unit is correct for describing the surface area of a hexagonal pyramid?
a. cm
b. cm squared
c. cm cubed

Answer: (b) Area is always “units squared”.

22. If r>0 and s <0, in which quadrant of the xy-plane does the point (r,s) lie?
a. Quadrant I
b. Quadrant II
c. Quadrant III
d. Quadrant IV

Answer: d. (+, -) in Quadrant IV

23. Which set of points is equidistant from the rays that form an angle? (Hint: Draw it)
a. Perpendicular bisector
b. Skew Line
c. Angle Bisector
d. Central Angle

Answer: C. Angle Bisector

24. Which unit is used to measure the amount of liquid needed to fill a cylindrical container?
a. inches
b. square inches
c. cubic inches

Answer: C. Filling a container is volume. Volume is always cubic units.