1
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?
(C) Show algebraically the relationship between the exterior angle BCD and angles A and B.
Ex:
(1) Find x and y(2) Find x
(3) Find x and y
(a) (b)
(c) (d)
(e) Find ?(f) Find x, y, z
(g)
Proofs
.
Given: ab, cd
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 / 24Interior 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 / 15ext angle / 6 / 8
interior angle / 165 / 178
Review Questions
Find:
mFHD = ______mGHB = ______mHDE = ______
mAHG = ______mBHC = ______mHDC = ______
mFHG= ______mDHC = ______mHCD = ______
Find:
m1 = ______m2 = ______m3 = ______
m4 = ______m5 = ______m6 = ______
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 mg = 81, me = 72, then mi = ______.
b) if mg = 46, mi = 79, then mh = ______.
10. For regular polygons, fill in the following chart completely.
Number of sides / 4Each 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: mABD = mAED
Prove: mC = mF
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?