1. 4.1 Apply Triangle Sum Properties

2. Objectives Identify and classify triangles by angles or sides
Apply the Angle Sum Theorem
Apply the Exterior Angle Theorem

3. Parts of a Triangle A triangle is a 3-sided polygon
The sides of ∆ABC are AB, BC, and AC
The vertices of ∆ABC are A, B, and C
Two sides sharing a common vertex are adjacent sides
The third side is called the opposite side
All sides can be adjacent or opposite (it just depends which vertex is being used) C A B
adjacent
Side opposite A

4. Classifying Triangles by Angles
One way to classify triangles is by their angles…
Acute all 3 angles are acute (measure < 90°)
Obtuse 1 angle is obtuse (measure > 90°)
Right 1 angle is right (measure = 90°)
An acute ∆ with all angles  is an equiangular ∆ .

5. Example 1:
ARCHITECTURE The triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLN as acute, equiangular, obtuse, or right.
60°
60°

6. Example 1:
Answer:
JMN has one angle with measure greater than 90, so it is an obtuse triangle.
JKO has one angle with measure equal to 90, so it is a right triangle.
OLN is an acute triangle with all angles congruent, so it is an equiangular triangle.

7. Classifying Triangles by Sides
Another way to classify triangles is by their sides…
Equilateral 3 congruent sides
Isosceles 2 congruent sides
Scalene no congruent sides

8. Example 2a: Identify the isosceles triangles in the figure if
Isosceles triangles have at least two sides congruent.
Answer: UTX and UVX are isosceles.

9. Example 2b: Identify the scalene triangles in the figure if
Scalene triangles have no congruent sides.
Answer: VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene.

10. Example 2c: Identify the indicated triangles in the figure.
a. isosceles triangles
Answer: ADE, ABE
b. scalene triangles
Answer: ABC, EBC, DEB, DCE, ADC, ABD

11. Example 3:
ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and
Since KLM is equilateral, each side has the same length. So
5 = d

12. Example 3: Next, substitute to find the length of each side. KL = 7
LM = 7
KM = 7
Answer: For KLM, and the measure of each side is 7.

13. Example 4:
COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides.

14. Example 4: Use the distance formula to find the lengths of each side.
Answer: ; since all 3 sides have different lengths, RST is scalene.

15. Exterior Angles and Triangles
An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ).
The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3). 1 2 3 4

16. Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°.
mX + mY + mZ = 180° X Y Z

17. Example 5: Find the missing angle measures.
Find first because the measure of two angles of the triangle are known.
Angle Sum Theorem
Simplify.
Subtract 117 from each side.

18. Example 5: Angle Sum Theorem Simplify. Subtract 142 from each side.
Answer:

19. Your Turn:
Find the missing angle measures.
Answer:

20. Theorem 4.2 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles m 1 = m 2 + m 3 1 2 3 4

21. Example 6: Find the measure of each numbered angle in the figure.
Exterior Angle Theorem
Simplify.
If 2 s form a linear pair, they are supplementary.
Substitution
Subtract 70 from each side.

22. Example 6: Exterior Angle Theorem Substitution
Subtract 64 from each side.
If 2 s form a linear pair, they are supplementary.
Substitution
Simplify.
Subtract 78 from each side.

23. Example 6: Angle Sum Theorem Substitution Simplify.
Subtract 143 from each side.
Answer:

24. Your Turn: Find the measure of each numbered angle in the figure.
Answer:

25. Corollaries
A corollary is a statement that can be easily proven using a theorem.
Corollary 4.1 – The acute s of a right ∆ are complementary.
Corollary 4.2 – There can be at most one right or obtuse  in a ∆.

26. Example 3:
GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.
Corollary 4.1
Substitution
Subtract 20 from each side.
Answer:

27. Your Turn:
The piece of quilt fabric is in the shape of a right triangle. Find if is 62.
Answer:

28. Assignment
Geometry: pg #1 – 10, 14 – 19,