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**Triangle Fundamentals**

Intro to G.10 Triangle Fundamentals Modified by Lisa Palen

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**Triangle What’s a polygon?**

Definition: A triangle is a three-sided polygon. What’s a polygon?

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Polygons Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons These figures are polygons

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**Definition of a Polygon**

A polygon is a closed figure in a plane formed by a finite number of segments that intersect only at their endpoints.

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**Triangles can be classified by:**

Their sides Scalene Isosceles Equilateral Their angles Acute Right Obtuse Equiangular

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**Classifying Triangles by Sides**

Scalene: A triangle in which no sides are congruent. BC = 5.16 cm B C A BC = 3.55 cm A B C AB = 3.47 cm AC = 3.47 cm AB = 3.02 cm AC = 3.15 cm Isosceles: A triangle in which at least 2 sides are congruent. HI = 3.70 cm G H I Equilateral: A triangle in which all 3 sides are congruent. GI = 3.70 cm GH = 3.70 cm

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**Classifying Triangles by Angles**

Obtuse: 108 44 28 B C A A triangle in which one angle is.... obtuse. Right: A triangle in which one angle is... right.

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**Classifying Triangles by Angles**

Acute: 57 47 76 G H I A triangle in which all three angles are.... acute. Equiangular: A triangle in which all three angles are... congruent.

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**Classification of Triangles Flow Charts Venn Diagrams**

with Flow Charts and Venn Diagrams

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**Classification by Sides**

polygons Polygon triangles Triangle scalene isosceles Scalene Isosceles equilateral Equilateral

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**Classification by Angles**

polygons Polygon triangles Triangle right acute equiangular Right Obtuse Acute obtuse Equiangular

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**Naming Triangles We name a triangle using its vertices.**

For example, we can call this triangle: ∆ABC ∆ACB Review: What is ABC? ∆BAC ∆BCA ∆CAB ∆CBA

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**Parts of Triangles Every triangle has three sides and three angles.**

For example, ∆ABC has Sides: Angles: CAB ABC ACB

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**Opposite Sides and Angles**

Side opposite of BAC : Side opposite of ABC : Side opposite of ACB : Opposite Angles: Angle opposite of : BAC Angle opposite of : ABC Angle opposite of : ACB

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**Interior Angle of a Triangle**

An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides. For example, ∆ABC has interior angles: ABC, BAC, BCA

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Exterior Angle An exterior angle of a triangle (or any polygon) is an angle that forms a linear pair with an interior angle. They are the angles outside the polygon formed by extending a side of the triangle (or polygon) into a ray. Exterior Angle Interior Angles A For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB. D B C

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**Interior and Exterior Angles**

The remote interior angles of a triangle (or any polygon) are the two interior angles that are “far away from” a given exterior angle. They are the angles that do not form a linear pair with a given exterior angle. For example, ∆ABC has exterior angle: ACD and remote interior angles A and B Exterior Angle Remote Interior Angles A D B C

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Triangle Theorems

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**m<A + m<B + m<C = 180**

Triangle Sum Theorem The sum of the measures of the interior angles in a triangle is 180˚. m<A + m<B + m<C = 180 IGO GeoGebra Applet

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Third Angle Corollary If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.

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**Third Angle Corollary Proof**

Given: The diagram Prove: C F statements reasons 1. A D, B E 2. mA = mD, mB = mE 3. mA + mB + m C = 180º mD + mE + m F = 180º 4. m C = 180º – m A – mB m F = 180º – m D – mE 5. m C = 180º – m D – mE 6. mC = mF 7. C F 1. Given 2. Definition: congruence 3. Triangle Sum Theorem Subtraction Property of Equality Property: Substitution Definition: congruence QED

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**Corollary Each angle in an equiangular triangle measures 60˚. 60 60**

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Corollary There can be at most one right or obtuse angle in a triangle. Example Triangles???

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Corollary Acute angles in a right triangle are complementary. Example

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**Exterior Angle Theorem**

The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Remote Interior Angles A Exterior Angle D Example: Find the mA. B C 3x - 22 = x + 80 3x – x = 2x = 102 x = 51 mA = x = 51°

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**Exterior Angle Theorem**

The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. GeoGebra Applet (Theorem 1)

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**Special Segments of Triangles**

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**Introduction There are four segments associated with triangles:**

Medians Altitudes Perpendicular Bisectors Angle Bisectors

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**Median - Special Segment of Triangle**

Definition: A segment from the vertex of the triangle to the midpoint of the opposite side. B A D E C F Since there are three vertices, there are three medians. In the figure C, E and F are the midpoints of the sides of the triangle.

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**Altitude - Special Segment of Triangle**

The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side. Definition: B A D F In a right triangle, two of the altitudes are the legs of the triangle. B A D F I K In an obtuse triangle, two of the altitudes are outside of the triangle.

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**Perpendicular Bisector – Special Segment of a triangle**

A line (or ray or segment) that is perpendicular to a segment at its midpoint. Definition: The perpendicular bisector does not have to start from a vertex! R O Q P Example: M L N C D A E A B B In the isosceles ∆POQ, is the perpendicular bisector. In the scalene ∆CDE, is the perpendicular bisector. In the right ∆MLN, is the perpendicular bisector.

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