4.2 Angles of Triangles

Objectives Apply the Angle Sum Theorem Use theorems, conjectures, definitions and properties to indicate WHY? Each step can be used.

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

Example 1: 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 1: WHY??? Angle Sum Theorem Simplify. Subtract 142 from each side. Answer:

Your Turn: Find the missing angle measures. Answer:

Theorem – Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangle are congruent. Abbreviation: If 2 s of one Δ are  to 2 s of another Δ, then third s are .

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

Example 2 (continued): Exterior Angle Theorem Substitution WHY??? 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 2 (continued) : Angle Sum Theorem Substitution Simplify. WHY??? 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 – The acute s of a right ∆ are complementary. Corollary – There can be at most one right or obtuse  in a ∆. Also called a Conjecture in our textbook.

Example 3: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. WHY??? Corollary – At most 1 rt triangle 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: