1. 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. 2 Ways to classify triangles
1) by their Angles
2) by their Sides

3. 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

4. 2) Sides Scalene Isosceles Equilateral - No sides congruent
- All sides are congruent

5. Parts of a Right Triangle
Hypotenuse
Side across the 90o angle. Always the largest in a right triangle
Leg
Leg
Sides touching the 90o angle

6. 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.

7. 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

8. 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 =
So……. Its SCALENE

9. Name the triangle by its angles and sides

10. Isosceles Triangle A C B Angle where the 2 congruent sides meet
Vertex- A
Legs – the congruent sides
Leg
Base Angles:
Congruent
Formed where the
base meets the leg C B
Base-
Non congruent side
Across from the vertex

11. Example
Triangle TAP is isosceles with angle P as the Vertex. TP = 14x -5 , TA = 6x + 11 , PA = 10x 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
6x + 11 T A
PA= 10(12) + 43 = 163
TA = 6(12) + 11 = 83

12. 1. 2.

13. Example
BCD is isosceles with BD as the base. Find the perimeter if BC = 12x-10,
BD = x+5
CD = 8x+6 C
12x-10
8x+6
Ans: 12x-10 = 8x+6
12(4)-10 38
8(4)+6 38
X = 4
base
Re-read the question, you need to find the perimeter D B
X+5
Perimeter = = 85
(4)+5 9
Final answer

14. Triangle Sum Thm
The sum of the measures of the interior angles of a triangle is 180o.
mA + mB+ mC=180o
= 180 A B C

15. 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
15x = 165
x = 11
3x +5
mA = 3(11) +5 = 38o
8x + 22
mW = 8(11)+22 = 110o
4x - 12 W
mE = 4(11)-12 = 32o E
Triangle AWE is obtuse

16. Example 2 Solve for x . Ans: (5x+24) + (5x+24) + (4x+6) = 180

17. The end