Geometry

Acute angles are < 900

Obtuse angles are > 900

Right angles are = 900

Supplementary angles total to 1800.

Complementary angles total to 900.

Types of angles

If two angles are complementary and one is 2 times greater than the other, what is the measure of the smaller angle and what type of angle is it?

X = smaller angle 2x = larger angle Equation: x + 2x = 90

3x = 90X = 30 and the angle is acute

Practice Problem

Vertical angles are congruent 1&4, 2&3, 5&8, 6&7

Alternate interior angles are congruent 3&6, 4&5

Alternate exterior angles are congruent 1&8, 2&7

Corresponding angles are congruent 1&5, 3&7, 2&6, 4&8

Same side interior angles are supplementary 3&5, 4&6

Rules of angles

1 2 3 4

5 6 7 8

Practice Problem

1 2

4 3 5

6

8 7

If <1 = 2x+3 and <5=x+7What is the value of x?

2x-3 = x+7X= 10

The sum of the angles of a triangle is 180°.

Isosceles triangle – 2 sides and base angles congruent

Equilateral triangle – all sides and angles congruent

The sum of the two remote interior angles = the value of the exterior angle

Rules of Triangles

In a triangle the second angle is 2 time the first angle. The third angle is 5 more than the second angle. Find the measure of each angle.

Practice Problem

x

2x

2x+5

X + 2x + (2x +5) = 1805x + 5 = 1807x =

Only works with a RIGHT triangle

SIDE2 + SIDE2 = HYPOTENUSE2

a2 + b2 = c2

But Implies: if a2 + b2 < c2 C is an acute angle (<90°) if a2 + b2 > c2 C is an obtuse angle (>90°)

Pythagorean Theorem

Parallelogram – ◦ opposite sides are parallel and congruent ◦ Opposite angles are =◦ The diagonals bisect each other

Rectangle – a parallelogram with right angles Square – a rectangle with all sides equal Trapezoid: has one pair of parallel sides

The Sum of the angles of a polygon = 180(n-2)

Rules of Other Shapes

What is the sum of the angles of a hexagon:

180(6-2)==720

Practice Problem

Triangle Congruences

Side-Side-Side Angle-Side-Angle

12

Triangle Congruences

Side-Angle-Side

When trying to prove that two triangles are congruent, use matching parts and figure out which congruence postulate to use!

If triangles are similar, the sides are in proportion and so are the perimeters

Similarity

x

x

x

3

1236912

4

12

4

x

2

6

9

r= radius d= Diameter

2r=d

Circumference: C = 2Πr

Area = A = Πr2

Circles contain 360°

Circles

1. Find the area of a circle with diameter = 12 “

A. 144Π B. 36 Π C. 12 Π D. 6 Π

The correct answer is B

Practice Problems for Circles

Central Angle – angle formed by two rays extending from the center

Central Angle

A

Sector of Area

C

Arc

360

Angle Central

Find the length of the arc intercepted by a 30 degree angle in a circle with radius = 4.

arc = (30 x 8Π) / 360 arc =

Practice Problem

8360

30 arc

3

2

r

arc

2360

30

Inscribed angles have their vertex on the circle and the intercepted are =

½ the measure of the angle

Inscribed – inside an object◦ The circle is inscribed by the square

It just touches the edges

Circumscribed – surrounding an object◦ The circle circumscribed the square

It just touches the edges

Terminology