Angles of a Triangle

Geo 10 Ch 3.4-3.5

..Angles of a Triangle

3-4

(A) A triangle is a figure formed by 3 ______joining 3 ______

Draw a picture and describe each of the following triangles.

1. Scalene4. Acute

2. Isoceles5. Obtuse

3. Equilateral6. Right

(B) Given triangle ABC. Draw a line throught B that is parallel to segment AC.

Show algebraically how the sum of the angles of the triangle equal 180 degrees.

EX. (1) Find x(2) Find x

(3) Find aWhat must be true about angle in a right triangle?

(4) Find x and yConclusion?

Ex:

(1) Find x and y(2) Find x

(3) Find x and y

(a) (b)

(e) Find ?(f) Find x, y, z

(g)

Proofs

Given: ab, cd
Prove: 1 7

3-5

The sum of the measures of the angles of a convex polygon with n sides

is ______

The sum of the measures of the exterior angles of any polygon, one angle at

each vertex, is ______.

What if the polygon is a regular polygon? What do you know about the angles?

Int angles = ______

Ext angles = ______

EXAMPLES

Find the interior sum, exterior sum, interior and exterior angles for each regular polygon with n sides.

(1)

Number of sides (n) / 6 / 10 / 24
Interior sum
Exterior sum
Interior angle
Exterior angle

Which is easier to find, an interior angle or an exterior angle?

2) An interior angle of a regular polygon = 160. How many side does it have?

3) Fill in the blanks

number of sides / 9 / 15
ext angle / 6 / 8
interior angle / 165 / 178

Review Questions

Find:

mFHD = ______mGHB = ______mHDE = ______

mAHG = ______mBHC = ______mHDC = ______

mFHG= ______mDHC = ______mHCD = ______

Find:

m1 = ______m2 = ______m3 = ______

m4 = ______m5 = ______m6 = ______

3)Find the sum of the measures of the marked angles.

Sum = ______

4) Find the measure of each interior angle of a regular 25-gon.

______

Review Cont.

5. If a polygon has 11 sides, find the sum of the interior angles.

6. If a regular polygon has 36 sides, find each interior angle.

7. If 7 exterior angles of an octagon each measure 42, what is the measure of the 8th angle?

8. The second angle of a triangle is three times the first, and the third angle is nine less than

five times the first. Find the measure of each angle of the triangle.

9. a) if mg = 81, me = 72, then mi = ______.

b) if mg = 46, mi = 79, then mh = ______.

10. For regular polygons, fill in the following chart completely.

Number of sides / 4
Each Interior  / 171 / 170
Each Exterior  / 3

Solve for the variable(s) in each drawing.

11. 12.

13. 14.

15A. *15B..

x = ______y = _____ z = _____x = _____ y = ______

Proofs:

16. Given: mABD = mAED

Prove: mC = mF

17. Given: , R T

Prove:

18. Given: m  n; 1 3

Prove: a  b

19. Given ,

Prove: 1 3

20. 21.

Given: , Given: <A <D

Prove: <1 <8Prove: <B <E

*22. Given : l || m, find x

23. Given: , ,

Prove:

*24 .

Given: , <1 <4

Prove:

*25. <1 = <2, marked lines are parallel find <3

26.

Given: bisects <ABC

Prove:

*27.

Given: <1 = <2

Prove: <4 = <ABC

Review Answers:

1. 12090120

606060

306060

2. 1= 26 2=26 3=64 4=385=52 6=64

3. NONE

4. 165.6

5.1620

6. 170

7. 66

8. 21, 63, 96

9. a) 153 b) 147

10. 44012036

90171177170

909310

11. x=43 y=47

12. x=y=70z=136

13. x=10y=20z=130

14. 50, 40, 110

15. x=90y=119z=29

15B. x=8, y=12

25. <3 = 140

CH 3.4-3.5

DEFINITIONS

1. EQUILATERAL TRIANGLE

2. REGULAR POLYGON

THEOREMS/COROLLARIES, ETC

THIRD ANGLE COROLLARY

ANGLES IN EQUILATERAL TRIANGLE

ACUTE ANGLES IN RIGHT TRIANGLE

EAT

THE SUM OF INTERIOR ANGLES

THE SUM OF EXTERIOR ANGLES

TO FIND EACH INTERIOR ANGLE IN REG . POLY

TO FIND EACH EXTERIOR ANGLE IN REG. POLY.

SUPPLEMENTARY PROBLEMS

1.Given two triangles where two angles in each are equal to each other. What can you say about the third angles?

2. Given the following diagram, <ABD is called an exterior angle of the triangle , being made up of one side of the triangle and an extension of another. How many exterior angles can you find at each vertex of the triangle?

(a) If the m<BAC = 50° and the m<ACB = 20°. Find the m<ABC and the m<ABD.

(b) If the m<BAC = x and m<ACB = y, find the m>ABD and come up for a rule about each of the exterior angles of a triangle?

3.The word polygon means “many angles”. Each polygon below is formed by coplanar segments, called sides, and we will denote each side as “n”.

Name each polygon and find the measure of all the angles.

Name ______

Sum ______

___ 180 = ___ 180 = ___ 180 = ___ 180

See the pattern?

What if I have a polygon with n sides, what is the sum of the interior angles? Give me a rule to follow for the sum of the interior angles of a polygon with n sides.

4. A polygon with equal sides and equal angles is called regular. How could you find each angle of a regular polygon?

5. In the diagram, find the sum of the exterior angles, one at each vertex.

6. Since the angles of a triangle add up to 180°,

a) make up a triangle and find the sum of the exterior angles, one at each vertex.

b) Make up a quadrilateral and find the sum of the exterior angles, one at each vertex.

c) Make up a hexagon and find the sum of the exterior angles, one at each vertex.

d) What have you observed?