Lesson 3.6 Overlapping Triangles

Transcription

1 Lesson 3.6 Overlapping Triangles Getting Ready: Each division in the given triangle is 1 unit long. Hence, the side of the largest triangle is 4- unit long. Figure Something to think about How many 1 unit triangles are there? 2-units triangles? 3-unit triangles? 4-unit triangles? Developing Skills: Sometimes a geometric figure contains two or more triangles that overlap each other. In such case, we need to untangle the figure to identify the triangles involved. Sometimes two triangles that we want to prove congruent have common parts with the two other triangles that can easily be proven congruent. In this case we may be able to use corresponding parts of these triangles to prove that the original triangles are congruent. Pre-Example 1: Name the triangles in the figure. Figure

2 Pre-Example 2: Name the pairs of triangles that can be proven congruent in the given figures. Figure Example 1: CA DA WA LA NA MA Example 2: Lesson 3.7 Theorems on Isosceles Triangles Theorem The Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite them are also congruent. Proof: ABC with AB AC Draw AD as an angle bisector of A intersecting BC at D B C

3 Corollary An equilateral triangle is also equiangular. Corollary An equilateral triangle has three 60 degrees angles. Theorem The Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite them are also congruent. Proof: B C Draw AD as an angle bisector of A intersecting BC at D AB AC Corollary An equiangular triangle is also equilateral. Let's Practice: Direction: Using theorems and corollaries about Isosceles, Equilateral and Equiangular triangles find the values of x

4 Example 4: m = n x = y A = B AD = BC Example 5: ABC is an isosceles with AC = BC BD is an angle bisector of ABC AE = DE DF is an angle bisector of ADE m EDA = 2m EDF m CBD = m EDF

5 Assignment: Direction: Complete the proof of the following OP OT UB US LE BS IR NA ULE is an isosceles triangle 3 4 Lesson 3.8 Measuring Angles in Triangles Theorem The Sum of the Measures of the Angles of a Triangle Theorem The Sum of the degree measures of the angles of a triangle is 180. Proof: ABC There is a line m through A parallel to BC m BAC + m B + m C = 180

6 Theorem The Third Angles Theorem If two angles of one triangle are congruent to two angles of another, then the third angles are congruent. Theorem If one angle of a triangle is right or obtuse, then the other two angles are acute. Theorem The acute angles of a right triangle are complementary. Lesson 3.9 Exterior and Interior Angles of a Triangle Getting Ready: If you extend one side of a triangle from one vertex, then you have constructed an exterior angle at that vertex. Figure Definition Exterior Angle It is an angle which forms a linear pair with an angle of the triangle. Definition Remote Interior Angles These are two angles in the triangle that do not have the same vertex as the exterior angle. Definition Adjacent Interior Angle For each exterior angle of a triangle, there corresponds an adjacent interior angle and a pair of remote interior angles. Theorem The Exterior Angle Theorem (EAT) The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

7 Proof: ABC m A + m B = m ACD Corollary The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angle. Assignment: Direction: Using the definitions, theorems and corollary in Lesson 3.9, find the value of x for the following given

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