4.1 Apply Triangle Sum Properties

Objectives

Identify and classify triangles by angles or sides

Apply the Angle Sum Theorem

Apply the Exterior Angle Theorem

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)

adjacent

adjacent

Side opposite A

C

A

B

Classifying Triangles by Angles

Obtuse1 angle is

obtuse (measure > 90°)

Right1 angle is right(measure = 90°)

One way to classify triangles is by their angles…

Acuteall 3 angles are

acute (measure < 90°)

An acute ∆ with all angles is an equiangular ∆ .

ARCHITECTURE The triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLN as acute, equiangular, obtuse, or right.

Example 1:

60° 60°

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.

Example 1:

Classifying Triangles by Sides

Isosceles 2 congruent

sides

Scaleneno congruent

sides

Another way to classify triangles is by their sides…

Equilateral3 congruent

sides

Answer: UTX and UVX are isosceles.

Identify the isosceles triangles in the figure if

Isosceles triangles have at least two sides congruent.

Example 2a:

Identify the scalene triangles in the figure if

Answer: VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene.

Scalene triangles have no congruent sides.

Example 2b:

Identify the indicated triangles in the figure.

a. isosceles triangles

b. scalene triangles

Answer: ABC, EBC, DEB, DCE, ADC, ABD

Answer: ADE, ABE

Example 2c:

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

Example 3:

Next, substitute to find the length of each side.

Answer: For KLM, and the measure of each side is 7.

Example 3:

KL = 7 LM = 7 KM = 7

COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides.

Example 4:

Answer: ; since all 3 sides have different lengths, RST is scalene.

Use the distance formula to find the lengths of each side.

Example 4:

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).

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Theorem 4.1 – Triangle Sum Theorem

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

mX + mY + mZ = 180°

X

Y Z

Find the missing angle measures.

Find first because the measure of two angles of the triangle are known.

Angle Sum TheoremSimplify.

Subtract 117 from each side.

Example 5:

Answer:

Angle Sum TheoremSimplify.

Subtract 142 from each side.

Example 5:

Find the missing angle measures.

Answer:

Your Turn:

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

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Find the measure of each numbered angle in the figure.

Exterior Angle TheoremSimplify.

SubstitutionSubtract 70 from each side.

If 2 s form a linear pair, they are supplementary.

Example 6:

Exterior Angle Theorem

Subtract 64 from each side.

Substitution

Subtract 78 from each side.

If 2 s form a linear pair, they are supplementary.

Substitution

Simplify.

Example 6:

Subtract 143 from each side.

Angle Sum Theorem

Substitution

Simplify.

Answer:

Example 6:

Find the measure of each numbered angle in the figure.

Answer:

Your Turn:

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 ∆.

Corollary 4.1

Substitution

Subtract 20 from each side.Answer:

GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.

Example 3:

Answer:

The piece of quilt fabric is in the shape of a right triangle. Find if is 62.

Your Turn:

Assignment

Pre-AP Geometry:Pgs. 221-224

#1 – 6, 14 – 19, 21 – 26, 31 – 37