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Types of Angles
Type
Right
Sketch
Description
90°
Acute
0° - 90°
Obtuse
90° - 180°
Straight
180°
Reflex
180° - 360°
Angle Relationships
X – Theorem (vertically opposite angles)
a°
b° b°
a°
Z – Theorem (alternate interior angles)
a°
b°
a°
b°
F – Theorem (corresponding angles)
a°
a°
C – Theorem (co-interior angles)
a°
b°
a° + b° = 180°
Complementary Angles
b°
a°
Two angles that have a sum of 90°. (a° + b° = 90°)
Supplementary Angles
a°
b°
Two angle that have a sum of 180°. (a° + b° = 180°)
Triangles: All angles add up to 180°.
Quadrilaterals: All angles add up to 360°.
Types of Triangles
According to sides:
Name
Sketch
Equilateral
Description
All 3 sides are equal length (and all 3 angles are 60°).
Isosceles
2 sides are congruent (and 2 angles are congruent).
Scalene
No sides or angles are congruent.
According to angles:
Name
Sketch
Acute
Description
All angles are less than 90°.
Obtuse
One angle is more than 90°.
Right
One angle is 90°.
Congruent Triangles
Two triangles are congruent if all angles and all side lengths of one triangle match all
angles and all corresponding side lengths of the other triangle.
F
ABC  DEF
A
B
C
D
E
Congruent → same shape
→ same size
Similar Triangles
The triangles are similar if one triangle is an enlargement or reduction of the other.
Similar triangles have 2 properties:
1) corresponding angles are equal
2) corresponding sides are proportional (have the same ratio)
P
Q
W
R
X
PQR ~ WXY
Y
PQ QR PR


WX
XY WY
Similar → same shape
→ not the same size
Area of Similar Triangles
If two similar triangles have the side ratio of 1:x then the ratio of their areas will be 1:x2
Pythagorean Theorem
What do you know?
 a2 + b2 = c2
 only works for right angle triangles
 need 2 sides to find the third
 “c” is the hypotenuse which is the longest side and the side opposite the right
angle
Vocabulary
hypotenuse
leg
leg
9  3 or -3 because 3*3=9 and -3*-3=9
Principal Square Root:
The positive root (3) is said to be the principal root.
Radicals
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289,
324, 361, 400
x * x = x2 so x2 is a perfect square
x2 * x2 = x4 so x4 is a perfect square
x3 * x3 = x6 so x6 is a perfect square
 if you have an even exponent it is a perfect square.
Terminology:
radical
9 3
square root
radicand
Examples:
1. 49  7
2.
4

9
4

9
3. 1.21  1.1
4. 1600  40
2
3
Take the square root of the numerator and denominator
Think 121  11
Think 16  4
Simplifying Radicals
48
 16  3
 16  3
Steps:
1. Find the largest perfect square that is a factor of the
number (radicand).
2. Write the radicand as the product of the perfect square.
3. Simplify the perfect square.
4 3
Adding and Subtracting Radicals
When adding and subtracting radicals the radicand must be the same ( like terms ) before you add or
subtract them.
Example 1:
Since the radicals are the same, just add the numbers in front. DO NOT add the radicands.
3 3 3 2 3
2 3
Example 2:
Since the radicals are not all the same they cannot all be added. In this case add only the like
radicals.
4 2 3 5 2 2 6 5
 2 2 9 5
Example 3:
If the radicals are different, check to see if any can be simplified. Sometimes after the radicals are
simplified they will be the same and can be added or subtracted.
12  27  2 8
 43  93  2 42
 2 3  3 3  22 2
 2 33 34 2
5 34 2
Multiplying and Dividing Radicals
When multiplying radicals, multiply the number in front of the radical and multiply the radicands.
Eg. 4 2  5 3  20 6
(simplify your final answer if you can)
When dividing radicals, divide the number in front of the radical and divide the radicands.
Eg.
10 6
2 3
5 2
(simplify your final answer if you can)