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) C A B adjacent Side opposite A

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 ∆ .

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°

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.

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

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

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.

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

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

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.

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

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

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

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

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.

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

Your Turn: Find the missing angle measures. Answer:

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

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.

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.

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

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

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 ∆.

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:

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

Assignment Geometry: pg 221-224 #1 – 10, 14 – 19, 21 - 37