Lines and Angles

PARALLEL LINES

• Def: line that do not intersect.

• Illustration:

• Notation: l || m AB || CD

lm

A

B

C

D

Examples of Parallel Lines• Hardwood Floor• Opposite sides of windows, desks, etc.• Parking slots in parking lot• Parallel Parking• Streets: Laramie & LeClaire

PERPENDICULAR LINES

• Def: Lines that intersect to form a right angle.

• Illustration:

• Notation: m n

• Key Fact: 4 right angles are formed.

m

n

Ex. of Perpendicular Lines• Window panes• Streets: Belmont and Cicero

AnglesTwo rays or line segments that meet at a point form an angle.

The point where the rays meet is called the vertex of the angle.

We measure the size of an angle using degrees.

Two ways to label angles:

1. by giving the angle a name, usually a lower-case letter like a or b, or sometimes a Greek letter like α (alpha) or θ (theta), or sometimes a number

2. or by the three letters on the shape that define the angle, with the middle letter being the vertex.

Example angle "a" is "BAC” (or“CAB”), and angle "θ" is "BCD” (or “DCB”)

Measuring Angles

• This is one degree:

• A Full Circle is 360°• Half a circle is 180°

(called a straight angle)• Quarter of a circle is 90°

(called a right angle)

Types of Angles Angle Names:Acute – less than 90 degreesRight – exactly 90 degrees (indicated on the GED by a square in the

corner of the angle)Obtuse – more than 90 degreesStraight – exactly 180 degrees

Complementary Angles

Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees.

One of the complementary angles is said to be the complement of the other.

• Supplementary angles add up to 180 degrees

Vertical Angles

Two straight lines crossing create vertical angles. Vertical angles (such as <BEC and <AED) have equal angle measurements.

In the diagram below, <AEB and <DEC are also vertical and therefore equal angles.

Corresponding Angles• A line going through two parallel lines creates corresponding

angles. • Corresponding angles (such as <D and <B) have equal

measurements. • In the diagram below, <C and <A are also corresponding and

therefore equal angles.

Corresponding

1234

5 6 7 8t

Corresponding

1234

5 6 7 8t

4 and 2 3 and 15 and 76 and 8

Angle Measurements

• If <VNL measures 130°, what is the measurement of <UNB?

Angle Measurements

• If <VNL measures 130°, what is the measurement of <UNB?

• <VNL and <UNB are vertical angles. Therefore, their measurements are the same. <UNB = 130°

Angle Measurements

• If <VNL measures 130°, what is the measurement of <LNB?

Angle Measurements

• If <VNL measures 130°, what is the measurement of <LNB?

• <VNL and <LNB are supplementary angles. • 180 – 130 = 50• <LNB = 50°

Angle Measurements

• If <SMP is a right angle and <ZMP measures 43°, what is the measurement of <ZMS?

Angle Measurements

• If <SMP is a right angle and <ZMP measures 43°, what is the measurement of <ZMS?

• <ZMP and <ZMS arecomplementary angles.90 – 43 = 47<ZMS = 47°

Angle Measurements

• <QFL is a straight angle. If <LFC measures 50°, what is the measurement of <QFC?

Angle Measurements

• <QFL is a straight angle. If <LFC measures 50°, what is the measurement of <QFC?

• The angles are supplementary• 180 – 50 = 130• <QFC = 130°

Angle Measurements

• If <AKS is 38°, find the measurements of all of the other angles

If <AKS is 38°, find the measurements of all of the other angles

• <EKR is vertical to <AKS,so <EKR = 38°

• <AKE is supplementary to <EKR, so <AKE = 142° (180 – 38)

• <RKQ is complementary to <EKR, so <RKQ = 52° (90 – 38)

• Finally, <SKQ is supplementary to <EKQ, so it must measure 90° (180 – 90)

Angle Measurements

• If <NTF measures 125°, what are the measurements of the other angles?

If <NTF measures 125°, what are the measurements of the other angles?

• <NTF is supplementary to <RTN,so <RTN must be 55° (180 – 125)

• <FTY is vertical to <RTN and supplementary to <NTF, so it must also be 55°

• <RTY is vertical to NTF and supplementary to <FTY, so it must also measure 125°

• <NTF corresponds with <KRT, so they have the same measurements. <KRT = 125°

• <TRU = 55° (corresponding to <FTY)• <XRU = 125° (corresponding to <RTY)• <XRK = 55° (corresponding to <RTN)

• WHEW!

• All the interior angles of any triangle will together add up to 180°

• All the interior angles of any quadrilateral (square, rectangle, parallelogram, trapezoid) will add up to 360°

• What is the measure of the missing angle, <BAC?

• What is the measure of the missing angle, <BAC?

All triangles have interior angles totaling 180°,So 180 – (52 + 48) gives us:<BAC = 80°

• Find the measure of the missing angle:

• Find the measure of the missing angle:

• All quadrilaterals haveinterior angles totaling 360°, so, 360 – (68+106+126) 360 – 300 = 60gives us:angle x measures 60°

Lines and Angles

• Pages 167 – 170 in the book– Check answers online

• Pages 29-30 in the GED Practice Packet– Enter answers online