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Geometry - Triangles - Document preview page 1

Geometry - Triangles - Page 1

Document preview content for Geometry - Triangles

Geometry - Triangles

This document provides study materials related to Geometry - Triangles. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

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Page 1 of 49
Geometry - Triangles - Page 1 preview imageStudy GuideGeometry–Triangles1. Angle Sum of a TriangleNow let’s explore one of the most important results in geometry.Using theParallel Postulate, we can prove a powerful theorem about triangles.Theorem 25: Angle Sum of a TriangleThe sum of the interior angles of any triangle is180°.In symbols:This is true forevery triangle—no matter its shape or size.•Small or large•Right, acute, or obtuse•Scalene, isosceles, or equilateralThe total of the three interior angles is always 180°.1.1Why This MattersThis theorem allows you to:•Find a missing angle in a triangle•Check whether angle measures form a valid triangle•Solve many geometry problems quicklyOnce you know two angles, you can always find the third.
Page 2 of 49
Geometry - Triangles - Page 2 preview imageStudy GuideExample 1If:•m∠A = 40°•m∠B = 60°Find m∠C.Step 1: Use the Triangle Sum TheoremStep 2: Substitute the Known ValuesStep 3: Combine What You KnowStep 4: Solve for the Missing AngleFinal Answer
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Geometry - Triangles - Page 3 preview imageStudy GuideSummaryIn any triangle:•Add the two known angles.•Subtract their sum from 180°.•The result is the missing angle.This rule is simple—but it’s one of the most useful tools in geometry.2. Exterior Angle of a TriangleNow let’s look at another important triangle idea:exterior angles.What Is an Exterior Angle?Figure 1 Exterior angle of a triangle.Anexterior angle of a triangleis formed when you extend one side of the triangle.Here’s what happens:•Start with a triangle.•Extend one side beyond a vertex.•The angle formed outside the triangle (next to an interior angle) is called anexterior angle.Important:It’s not the straight angle formed by the extension. It’s the angle outside the triangle that isadjacentto an interior angle.In the diagram,∠BCD is an exterior angle of△ABC.
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Geometry - Triangles - Page 4 preview imageStudy Guide2.1Why Exterior Angles MatterExterior angles are directly connected to the interior angles of the triangle.We already know:We also know that∠3 and∠4 form a straight line, so:Using these facts, we can prove something powerful.2.2Theorem 26: Exterior Angle TheoremAn exterior angle of a triangle is equal to the sum of the tworemote (nonadjacent) interior angles.In simple terms:The “remote” interior angles are the two inside angles that arenot next tothe exterior angle.So in the diagram:2.3Why This Is UsefulThis theorem lets you:•Find missing angles quickly•Solve triangle problems faster•Avoid adding all three interior angles every timeYou only need the two remote interior angles!
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Geometry - Triangles - Page 5 preview imageStudy GuideExample 1:SolveIn the figure:•m∠1 = 30°•m∠2 = 100°Find m∠4.Step 1: Use the Exterior Angle TheoremStep 2: Substitute the ValuesStep 3: AddFinal AnswerSummaryWhenever you see an exterior angle of a triangle:•Look at the two interior angles that are not next to it.•Add them together.•That sum equals the exterior angle.This is one of the most helpful shortcuts in triangle geometry!
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Geometry - Triangles - Page 6 preview imageStudy Guide3. Classifying Triangles by Sides or AnglesTriangles can be grouped in two different ways:1.By their sides2.By their anglesSome triangles may fit into one category from each group.For example, a triangle could be bothisosceles(by sides) andacute(by angles).Let’s look at each type clearly and simply.3.1Classifying Triangles by Their SidesThis method looks at how many sides are equal.1. Equilateral TriangleFigure 1Equilateral triangleAnequilateral trianglehasall three sides equalin length.In diagrams, small slash marks on the sides show they are equal.Because all sides are equal, something else is true:All the angles are equal too.
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Geometry - Triangles - Page 7 preview imageStudy Guide2. Isosceles TriangleAnisosceles trianglehasat least two sides equalin length.Figure 2 Isosceles trianglesThat means:•It could have exactly two equal sides.•Or it could have three equal sides (which would also make it equilateral).In most problems, when you see matching marks on two sides, you’re looking at an isosceles triangle.3. Scalene TriangleFigure 3 Scalene triangleAscalene trianglehasall three sides differentlengths.No sides are equal.This also means none of the angles are equal.
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Geometry - Triangles - Page 8 preview imageStudy Guide3.2Classifying Triangles by Their AnglesNow let’s look at triangles based on their angle measures.1. Right TriangleFigure 4 Right triangleAright trianglehasone right angle(90°).You’ll usually see a small square marking the 90° angle in a diagram.Only one angle can be 90° in a triangle.2. Obtuse TriangleFigure 5 Obtuse triangle
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Geometry - Triangles - Page 9 preview imageStudy GuideAnobtuse trianglehasone obtuse angle.An obtuse angle is:•Greater than 90°•Less than 180°Only one angle in a triangle can be obtuse.3. Acute TriangleFigure 6 Acute triangle.Anacute trianglehasall three angles less than 90°.Every angle is acute.4. Equiangular TriangleFigure 7 Equiangular triangleAnequiangular trianglehasall three angles equal.Since the sum of a triangle’s angles is 180°, we can figure out something important.
Page 10 of 49
Geometry - Triangles - Page 10 preview imageStudy Guide3.3Theorem 27Each angle of an equiangular triangle measures60°.Why?Because:So if all three angles are equal, each one must be 60°.Helpful ConnectionsHere’s an important relationship:•Everyequilateral triangleis alsoequiangular.•Everyequiangular triangleis alsoequilateral.That means:If all sides are equal → all angles are 60°.If all angles are 60° → all sides are equal.SummaryBy Sides:•Equilateral→ 3 equal sides•Isosceles→ at least 2 equal sides•Scalene→ no equal sidesBy Angles:•Right→ one 90° angle•Obtuse→ one angle greater than 90°•Acute→ all angles less than 90°•Equiangular→ all angles equal (each 60°)
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Document Details

University
Texas A&M University
Course
Geometry
Subject
Mathematics

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