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1-4 & 1-5 Angles Measures and Relationships

Objectives:

The student will be able to:

1. Measure and classify angles.2. Use congruent angles and the bisector of an

angle.3. Identify and use special pairs of angles.4. Identify perpendicular lines.

Classifying Angles

Acute Angles < 90° Right Angles = 90° Obtuse Angles > 90°

When naming angles using 3 letters, the vertex must be the second of the 3 letters. You can name an angle using a single letter only when there is exactly one angle located at the vertex.

Naming Angles:

Naming angles.

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In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of x.

A ray that divides an angle into two congruent angles. PQS ≅ TQS The bisector of PQT is QS .

Congruent Angles & Angle Bisector:

50 = 4x + 14-14 -14

36 = 4x9 = x

In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of .

Example:

6x - 2 = 3x + 13+ 2

3x = 15x = 5

-3x

Did we answer the question? NO!

PQTTQSPQS

6(5) – 2 + 3(5) + 13 = PQT

30 – 2 + 15 + 13 = PQT 56° = PQT

If and , find the value of .

Special Angle pairs

Adjacent Angles:

Vertical Angles:

Linear Pair: 1 4, 2 3, 5 8, 6 7

Two angles that are opposite angles. Vertical angles are congruent.

1 & 2 , 2 & 4 , 4 &3, 3 & 1,

5 & 6, 6 & 8, 8 & 7, 7 & 5

Supplementary angles that form a line (sum = 180)

1 23 4

5 6

7 8

Two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. 1 & 2, 1 & 3, 2 & 4, 3 & 4, 5 & 6, 5 & 7, 6 & 8, 7 & 8

Special Angle pairs

Congruent Angles:

Perpendicular angles:

Two or more angles that have the same measure.

Lines, segments, and rays that form right angles (90 degrees).

AEB & BEC, CED & DEA, AEB & DEC, BEC & AED

E

AEC & BED

Complementary & Supplementary Angles

Complementary Angles:

Supplementary Angles:

Two angles whose measures have a sum of 90°.

Two angles whose measures have a sum of 180°.

A + B = 30 + 60 = 90

F + G = 120 + 60 = 180

Identify:

Two Obtuse vertical angles:

Two acute adjacent angles:

An angle supplementary to TNU:

Find x so that .

If the two angles are perpendicular they MUST = 90° .

(9x + 5) + (3x + 1) = 90

12x + 6 = 90- 6 -6

12x = 84 x = 7

= 18021

x + (x – 18)= 180

2x – 18 = 180+18 +18

2x = 198x = 99

Example:

Find the measures of 2 supplementary angles if the difference in their measures is 18.

Are we through? NO!!

If x = 99, what are the measures of the supplementary angles?

99 99 -18 = 81

99 + 81 = 180

How can I check to see if that’s correct?

Find x and y so that KO and HM are perpendicular.

(3x + 6) + (9x) = 90

12x + 6 = 90- 6 -6

12x = 84x = 7

1. Find x by setting the two angles equal to 90.

90

2. Vertical angels tell us if , then .3. Find y by setting .

(3y + 6) = 90- 6 -6

3y = 84y = 28

1. Are the angles congruent? Yes – set the expressions equal to each other.

A = B

2. Do the angles add up to 90°?Yes – add the expressions and set them equal to 90°.

A + B = 90

3. Do the angles add up to 180°?Yes – add the expressions and set them equal to 180°.

A + B = 180

4. Do the angles add up to some other value given in the problem?

Yes – add the expressions and set them equal to the value.

A + B = other value