2.1:a Prove Theorems about Triangles

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

GSE’s

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.

CCSS

2 Ways to classify triangles

1) by their Angles

2) by their Sides

1)Angles

• Acute-

• Obtuse-

• Right-

• Equiangular-

all 3 angles less than 90o

one angle greater than 90o, less than 180o

One angle = 90o

All 3 angles are congruent

2) Sides

• Scalene

• Isosceles

• Equilateral

- No sides congruent

-2 sides congruent

- All sides are congruent

Parts of a Right Triangle

Leg

Leg

Hypotenuse

Sides touching the 90o angle

Side across the 90 o angle.

Always the largest in a right

triangle

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. Ans: DA = AR = DR =

116

80

52

116

80

52

So……. Its SCALENE

Name the triangle by its angles and sides

Legs – the congruent sides

Isosceles Triangle

A

B C

Leg

Base- Non congruent side Across from the vertex

Vertex- Angle where the 2 congruent sides meet

Base Angles:

•Congruent•Formed where the base meets the leg

ExampleTriangle TAP is isosceles with angle P as the Vertex. TP = 14x -5 , TA = 6x + 11 , PA = 10x + 43. Is this triangle also equilateral?

T A

P

14x-5

6x + 11

10x + 43

TP PA

14x – 5 = 10x + 43

4x = 48

X = 12

TP = 14(12) -5 = 163

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

CD = 8x+6

B

C

D

base

12x-10 8x+6

X+5

Ans: 12x-10 = 8x+6

X = 4

Re-read the question, you need to find the perimeter

12(4)-10

38

8(4)+6

38

(4)+5

9

Perimeter =38 + 38 + 9 = 85Final answer

Triangle Sum Thm

• The sum of the measures of the interior angles of a triangle is 180o.

• mA + mB+ mC=180o

+ + = 180

A

B C

Example 1

• Name Triangle AWE by its angles

4x - 128x + 22

3x +5

A

WE

(3x+5) + ( 8x+22) + (4x-12) = 180mA + mW+ mE=180o

15x + 15 = 180

15x = 165

x = 11

mA = 3(11) +5 = 38o

mW = 8(11)+22 = 110o

mE = 4(11)-12 = 32o

Triangle AWE is obtuse

Example 2

Solve for x .

5x +24

Ans: (5x+24) + (5x+24) + (4x+6) = 180

5x +24 + 5x+ 24 + 4x+6 = 180

14x + 54 = 180

14x = 126

x = 9

The end