Transcript PowerPoint Notes - Property of triangles

2.1:a Prove Theorems about
Triangles
CCSS
G-CO.10
Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to the
third side and half the length; the medians of a triangle meet at a point.
GSE’s
M(G&M)–10–2 Makes and defends conjectures, constructs
geometric arguments, uses geometric properties, or uses theorems
to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across
disciplines or contexts
2 Ways to classify triangles
1) by their Angles
2) by their Sides
1)Angles
o
all
3
angles
less
than
90
• Acute-
• Obtuse- one angle greater than 90o, less
than 180o
• Right- One angle = 90o
• Equiangular- All 3 angles are congruent
2) Sides
• Scalene
- No sides congruent
• Isosceles -2 sides congruent
• Equilateral - All sides are congruent
Parts of a Right Triangle
Leg
Leg
Sides touching
the 90o angle
Converse of the Pythagorean Theorem
Where c is chosen to be the longest of the three
sides:
If a2 + b2 = c2, then the triangle is right.
If a2 + b2 > c2 , then the triangle is acute.
If a2 + b2 < c2, then the triangle is obtuse.
Example of the converse
• Name the following triangles according
to their angles
1) 4in , 8in, 9 in
2) 5 in , 12 in , 13 in
4) 10 in, 11in, 12 in
Example on the coordinate plane
• Given  DAR with vertices D(1,6)
A (5,-4)
R (-3, 0)
Classify the triangle based on its
sides and angles.
52
Ans: DA = 116
116
AR = 80
DR = 52
80
So……. Its SCALENE
Name the triangle by its angles and sides
Isosceles Triangle
Vertex- Angle where the
2 congruent sides meet
A
Legs – the
congruent sides
Leg
Base Angles:
•Congruent
•Formed where the
base meets the leg
B
C
Base- Non congruent side
Across from the vertex
Example
Triangle TAP is isosceles with angle P as
the Vertex. TP = 14x -5 , TA = 6x + 11 ,
PA = 10x + 43. Is this triangle also
equilateral?
P
TP
 PA
14x – 5 = 10x + 43
14x-5
10x + 43
4x = 48
X = 12
TP = 14(12) -5 = 163
T
6x + 11
A
PA= 10(12) + 43 = 163
TA = 6(12) + 11 = 83
1.
2.
Example
•  BCD is isosceles with BD as the base.
Find the perimeter if BC = 12x-10,
BD = x+5
C
CD = 8x+6
12x-10
12(4)-10
8x+6
8(4)+6
38
38
base
B
D
X+5
(4)+5
9
Ans: 12x-10 = 8x+6
X=4
Re-read the question, you need to find the perimeter
Perimeter =38 + 38 + 9 = 85
Final
answer
Triangle Sum Thm
• The sum of the measures
of the interior angles of a
triangle is 180o.
A
• mA + mB+ mC=180o
+
+
= 180
B
C
Example 1
• Name Triangle AWE by its angles
mA + mW+ mE=180o
(3x+5) + ( 8x+22) + (4x-12) = 180
A
15x + 15 = 180
3x +5
15x = 165
x = 11
mA = 3(11) +5 = 38
8x + 22
o
mW = 8(11)+22 = 110o
mE = 4(11)-12 = 32o
W
E
Triangle AWE is obtuse
Example 2
Solve for x .
Ans: (5x+24) + (5x+24) + (4x+6) = 180
5x +24
5x +24 + 5x+ 24 + 4x+6 = 180
14x + 54 = 180
14x = 126
x=9
The end