Angle Relationships and Similar Trianglesmrspolandsmath.weebly.com/uploads/3/3/1/0/3310645/trig1.1-1.2.pdf · Angle Relationships and Similar Triangles Geometric Properties Triangles

Angles1.1Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles

Mrs. PolandJanuary 23, 2013

Angle Relationships and Similar Triangles

Geometric Properties ▪ Triangles

1.2

Friday, February 1, 2013

ObjectivesObjective #1: Students will be able to use vocabulary to describe situations

and answer questions.

Objective #2: Students will be able to express angles in decimal degrees and degrees, minutes and seconds.

Success criterion #1: I can use vocabulary to describe situations and answers.

Success criterion #4: I can use geometric properties of triangles to solve problems.

Objective #3: Students will be able to use geometric relationships to solve problems.

Success criterion #2: I can express angle measures in appropriate units.

Success criterion #3: I can use geometric properties of lines and angles to solve problems.


✤ An angle’s measure is generated by a rotation about the vertex.

✤ The ray in its initial position is called the initial side of the angle.

✤ The ray in its location after the rotation is the terminal side of the angle.

Basic Terminology


✤ Positive angle: The rotation of the terminal side of an

angle is counterclockwise.

Negative angle: The rotation of the terminal

side is clockwise.

Basic Terminology


✤ The most common unit for measuring angles is the degree.

Measuring Angles

A complete rotation of a ray gives an angle whose

measure is 360°.

of complete rotation gives an angle whose measure is 1°.


✤ Find the measure of each marked angle.

Example 1a FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Since the two angles form a right angle, they are complementary.

Combine terms.Divide by 9.

Determine the measure of each angle by substituting 10 for x:


✤ Find the measure of each marked angle.

Example 1b FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES (continued)

Since the two angles form a straight angle, they are supplementary.

The angle measures are and .


Degrees, Minutes, Seconds

✤ One minute is 1/60 of a degree.

One second is 1/60 of a minute.


✤ Convert 74 8’14” to decimal degrees

✤ Convert 34.817 to degrees, minutes and seconds.

Example 2Converting between decimal and degrees, minutes

and seconds

o

74 8’14”o 860+o= 74 + 14_

3600o=74 + .1333 + .0039 = 74.137o

34.817o .817+o= 34.817(60)+o= 34

o= 34 + 49.02’o= 34 + 49’+ .02(60)”o= 34 + 49’ + 1.2”


Example 3 Finding the sum and difference

(a)

Add degrees and minutes separately.

(b)

Write 90° as 89°60ʹ′.

Find the sum or difference.


Standard Position

✤ An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.


Quadrantal Angles

✤ Angles in standard position with their terminal sides along the x-axis or y-axis, such as angles with measures 90°, 180°, 270°, and so on.


Coterminal Angles

✤ Coterminal angles are angles that share the same initial and terminal sides and differ by a multiple of 360 degrees.


Example 4 FINDING MEASURES OF COTERMINAL ANGLES

Find the angle of least possible positive measure coterminal with an angle of 908°.

Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°.

An angle of 908° is coterminal with an angle of 188°.


Example 4 FINDING MEASURES OF COTERMINAL ANGLES (continued)

Find the angle of least possible positive measure coterminal with an angle of –75°.

Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°.

An angle of –75 ° is coterminal with an angle of 285°.


Example 4 FINDING MEASURES OF COTERMINAL ANGLES (continued)

Find the angle of least possible positive measure coterminal with an angle of –800°.

An angle of –800° is coterminal with an angle of 280°.

The least integer multiple of 360° greater than 800° is


Coterminal Angles

✤ To find an expression that will generate all angles coterminal with a given angle, add integer multiples of 360° to the given angle.

For example, the expression for all angles coterminal with 60° is


Example 5 FINDING ANGLE MEASURES

Angles 1 and 4 are alternate exterior angles, so they are equal.

Subtract 3x.Add 40.

Divide by 2.

Angle 1 has measure

Substitute 21 for x.

✤ Find the measure of angles 1, 2, 3, and 4, given that lines m and n are parallel.


Example 5 FINDING ANGLE MEASURES (continued)

Angle 4 has measureSubstitute 21

for x.

Angle 2 is the supplement of a 65° angle, so it has

measure .

Angle 3 is a vertical angle to angle 1, so its measure is 65°.


Example 6 APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY

The measures of two of the angles of a triangle are 48° and 61°. Find the measure of the third angle, x.

The third angle of the triangle measures 71°.

The sum of the angles is 180°.

Add.

Subtract 109°.


Example 7 FINDING ANGLE MEASURES IN SIMILAR TRIANGLES

In the figure, triangles ABC and NMP are similar.Find the measures of angles B and C.

Since the triangles are similar, corresponding angles have the same measure.

B corresponds to M, so angle B measures 31°.

C corresponds to P, so angle C measures 104°.


Example 8 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES

In the figure, triangles ABC and DFE are similar.Find the measures of sides DF and EF.

Since the triangles are similar, corresponding sides are proportional.

DF corresponds to AB, and DE corresponds to AC, so

Side DF has length 12.

EF corresponds to CB, so

Side EF has length 16.Friday, February 1, 2013

Example 9 FINDING THE HEIGHT OF A FLAGPOLE

Firefighters at a station need to measure the height of the station flagpole. They find that at the instant when the shadow of the station is

18 m long, the shadow of the flagpole is 99 ft long. The station is 10 m high. Find the height of the flagpole.

Since the two triangles are similar, corresponding sides are proportional.

Lowest terms

The flagpole is 55 feet high.


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Angle Relationships and Similar Trianglesmrspolandsmath.weebly.com/uploads/3/3/1/0/3310645/trig1.1-1.2.pdf · Angle Relationships and Similar Triangles Geometric Properties Triangles