Classifying Triangles

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Presentation transcript:

Classifying Triangles Angles of Triangles Congruent Triangles

Classifications of Triangles

Examples Find the measures of the sides of isosceles triangle ABC.

Examples Find the measures of the sides of isosceles triangle ABC. 4x + 1 = 5x – 0.5 1.5 = x Substitute for x

Angles of Triangles Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180.

Auxiliary Line An auxiliary line is an extra line or segment drawn in a figure to help analyze geometric relationships.

Examples Find the measures of the numbered angles.

Examples ∠1 + 57 = 180 ∠1 = 123 57 + 71 + ∠2 = 180 128 + ∠2 = 180 ∠2 = 52 ∠1 + 28 + ∠3 = 180 123 + 28 + ∠3 = 180 ∠3 = 29

Exterior Angle Theorem An angle formed outside of a triangle by one side of the triangle and the extension of an adjacent side. Each exterior angle of a triangle has two remote interior angles that are not adjacent to the exterior angle

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.

Corollaries A corollary is a theorem with a proof that follows as a direct result of another theorem.

Triangle Angle-Sum Corollaries The acute angles of a right triangle are complementary.

Triangle Angle-Sum Corollaries There can be at most one right or obtuse angle in a triangle.

Congruence Two polygons are congruent if and only if corresponding parts are congruent. Corresponding parts include corresponding angles and corresponding sides.

Congruence

Congruence Statement

Examples Identify all congruent corresponding parts, then write a congruence statement.

Examples ∠J ≅ ∠P ∠K ≅ ∠M ∠L ≅ ∠Q JK ≅ MP KL ≅ MQ LJ ≅ QP

CPCTC Corresponding Parts of Congruent Triangles are Congruent

Examples In the diagram, △ABC ≅ △DFE. Find the values of x and y.

Examples In the diagram, △ABC ≅ △DFE. Find the values of x and y. B ≅ F, 8y - 5 = 99 y = 13 BC ≅ FE, 2y + x = 38.4 2(13) + x = 38.4 x = 12.4

Third Angles Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Why? Why would it be important to know that things are congruent? When would this occur in the real world?