Focus – Prove the Triangle Sum Theorem

Triangle Sum Theorem

Focus - Prove the Triangle Sum Theorem

Focus – Prove the Triangle Sum Theorem

If you add up the measures of the interior angles of a

triangle, What is the total?

180º

25º

65º

Focus – Prove the Triangle Sum Theorem

How many lines can be drawn through point B that would

become parallel to M�������������� �

M

B

Focus – Prove the Triangle Sum Theorem

Unique Parallel line Postulate – Through a point not on a given line, there is one and only one line parallel to the

given line.

m

b

This postulate will be used in the proofon the next slide.

Focus – Prove the Triangle Sum Theorem

Given:Prove: 1. ABC is a triangle2. Draw helping line through B so that

3.

4.5.6.7.

1. Given2. Unique Parallel Postulate

(pg. 439)3. If the sides of an angle form

a straight line, then the angle is a straight angle with a measure of 180º

4. Whole-Parts Postulate5. Substitution6. Alternate Interior Angle

Theorem7. Substitution Property

ABC1 2 3 180m m m

||DE AC���������������������������������������� ���DE�������������� �

B

CA

2

1 3

D

180m DBE

4 2 5m DBE m m m

E4 5

4 2 5 180m m m

1 4; 3 5m m m m

1 2 3 180m m m

Focus – Prove the Triangle Sum Theorem

Quadrilateral Theorem

• The sum of the measures of the angles of a quadrilateral are:

• What is the sum of the measures of a rectangle, which is a quadrilateral?

360

Focus – Prove the Triangle Sum Theorem

Theorem:Opposite Angles of

QuadrilateralsIf both pairs of opposite angles of a

quadrilateral are equal in measure, …

then the quadrilateral is a parallelogram.

Focus – Prove the Triangle Sum Theorem

What can we find out?

• The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?

75

Focus – Prove the Triangle Sum Theorem

What can we find out?

• The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?

75

75 90 180

165 180

15

x

x

x

15

Focus – Prove the Triangle Sum Theorem

Find the missing angle.

47º

Focus – Prove the Triangle Sum Theorem

Find the missing angle.

47º

43

Focus – Prove the Triangle Sum Theorem

Find the missing angles.

120º

xº 55º 25º yº

wº

120x

35y

35w

Order: y w x

Focus – Prove the Triangle Sum Theorem

Exterior Angle Theorem• An exterior angle of a triangle…• … is equal in measure to the sum of

the measures of its two remote interior angles.

Focus – Prove the Triangle Sum Theorem

Find the missing angles’ measurements (L and J).

120º

55º 25º

12035 35J L

90 55L 120 35J

remote interior angles

Focus – Prove the Triangle Sum Theorem

What is the measure of Angle U?

120º

55º 25º

12035 35J L

145L 155J

60

?

T

U

T U

To find the measure of angle T, use the definitionof supplemental angles.To find the measure of angle U, use the verticalangle theorem.

Focus – Prove the Triangle Sum Theorem

In today’s lesson we…

• Proved the Triangle Sum Theorem (180 degrees)

• Found Quadrilaterals’ Interior Angles Theorem to state the total degrees is 360.

• Used the above theorems and the Exterior Angle Theorem to find measurements of a Triangle’s exterior angle.

Focus – Prove the Triangle Sum Theorem

Assignment: pg. 443: 2-9, 13

Focus – Prove the Triangle Sum Theorem

For Warmup

70°

y°

z° w°

D E50° x°

(x-30)°

x°

70°

60°

50° 110°

D E50° 70°

(80-30)°

80°

Focus – Prove the Triangle Sum Theorem

From tdn.com last night.OLYMPIA, Wash. – The state Senate has made good on a

promise to help students graduate from high school by passing a bill that would eliminate one diploma obstacle.

The measure would eliminate the requirement to keep taking the math Washington Assessment of Student Learning every year until graduation.

High school students who fail the 10th grade math test, would still be required to keep earning math credits, however.

The bill, which passed on a vote of 43-4, still requires approval in the House. It would affect the class of 2009, if it passes both houses and is signed by the governor.

• The bill is Senate Bill 5498.• * Legislature (weblink) <http://www.leg.wa.gov>

Focus – Prove the Triangle Sum Theorem

Want to spend class time preparing for math WASL

questions?• Group Work• Group Points• Three sections will vote.• 60% will show favorable response.

Focus – Prove the Triangle Sum Theorem

Pg. 446: #17 Proof of Theorem

If both pairs of opposite angles of a quadrilateral are equal in measure, then the

quadrilateral is a parallelogram.

Focus – Prove the Triangle Sum Theorem

Given: JKLM is a quadrilateral

Prove: JKLM is a parallelogram 1. JKLM is a quadrilateral2. 3.4.5.6.7. are supplementary

angles

8.9.10. are

supplementary angles

11.12. JKLM is a

parallelogram

1. Given2. Quadrilateral Sum Theor.3. Given4. Substitution5. Distributive Property6. Division Property of Equality7. Def. of Supplementary

angles8. Converse of Co-Interior Th.9. Substitution10. Def. of Supplemental

Angles11. Chuck Taylor Co-Interior Th.

12. Def. of Parallelogram

360m J m K m L m M

M

;m J m L m K m M

360m J m M m J m M 2( ) 360m J m M

180m J m M &m J m M

m J m L

J

L

K

m K m M

||JK ML

&m J m K

180m J m K

||JM KL

statementsjustifications

Focus – Prove the Triangle Sum Theorem

Pg. 447: #34 Proof of Theorem

An Exterior Angle of a Triangle is equal in measure to the sum of its two remote

interior angles.

Focus – Prove the Triangle Sum Theorem

Given: with exterior Prove: 1. ABC is a triangle

with exterior angle 4.

2.

3.4.5.6.7.

1. Given

2. Straight Angle Postulate

3. Whole-Parts Postulate4. Substitution5. Triangle Sum

Theorem6. Substitution Property7. Subtraction Prop. Of

Equality

ABC4 1 2m m m

180m BCD

C D

3 4m BCD m m

3 4 180m m

1 2 3 180m m m

3 4 1 2 3m m m m m

4 1 2m m m

4A

B2

1

3 4

statements justifications

Focus – Prove the Triangle Sum Theorem

Assignment: pg. 444-448: 14-18a, 26-34, 37-40