C. N. Colon Geometry St. Barnabas HS. Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details,

Slides:



Advertisements
Similar presentations
Chapter 4a: Congruent Triangles By: Nate Hungate, Gary Russell, J. P
Advertisements

Proving Triangles Congruent Geometry D – Chapter 4.4.
1 CPCTC SIDE-ANGLE-SIDE ANGLE-ANGLE-SIDE PROBLEM 1 SIDE-SIDE-SIDE PROBLEM 3 ANGLE-SIDE-ANGLE Standards 4 and 5 SUMMARY: CONGRUENCE IN TRIANGLES SUMMARY:
Chapter 4: Congruent Triangles
Geometry 1 Unit 4 Congruent Triangles
2.3: Exploring Congruent Triangles
Congruent Polygons Have congruent corresponding parts. Have congruent corresponding parts. When naming congruent polygons, always list corresponding vertices.
Congruent Triangles Geometry Chapter 4.
Chapter 4 Congruent Triangles.
Chapter 4: Congruent Triangles
CHAPTER 4 Congruent Triangles SECTION 4-1 Congruent Figures.
$100 $200 $300 $400 $500 $200 $300 $400 $500 Classifying Triangles Proving Congruence Coordinate Proof Congruence in Right Triangles Isosceles Triangles.
Proving Triangles Congruent Advanced Geometry Triangle Congruence Lesson 2.
Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles.
Congruent Triangles.  Standard 25: Identify interior and exterior angles of a triangle and identify the relationships between them.  Standard 26: Utilize.
Section 4.1 Congruent Polygons. Polygons Examples of Polygons Polygons Examples of Non-Polygons Non-Polygons.
Introduction Think of all the different kinds of triangles you can create. What are the similarities among the triangles? What are the differences? Are.
6.0 Geometric Proofs Math 10Geometry Unit Lesson 1 Lesson 1.
Geometry – Chapter 4 Congruent Triangles.
4.6 Isosceles, Equilateral, and Right Triangles Geometry Mrs. Spitz Fall 2009.
EXAMPLE 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or.
2.3: Exploring Congruent Triangles M(G&M)–10–4 Applies the concepts of congruency by solving problems on or off a coordinate plane; or solves problems.
Proving Triangles Congruent. Warm Up Objectives Can you prove triangles congruent using SSS, SAS, ASA, AAS, and HL?
& 5.2: Proving Triangles Congruent
4.1 Classifying Triangles
Adapted from Walch Education Isosceles triangles have at least two congruent sides, called legs. The angle created by the intersection of the legs is.
Fall 2012 Geometry Exam Review. Chapter 1-5 Review p ProblemsAnswers 1One 2a.Yes, skew b.No 3If you enjoy winter weather, then you are a member.
4.1: Apply Triangle Sum Properties
4.5 Isosceles and Equilateral Triangles. Isosceles Triangles At least two sides are of equal length. It also has two congruent angles. Base Angles Base.
Chapter 4 Congruent Triangles In this chapter, you will: classify triangles by their parts, apply the Angle Sum Theorem and the Exterior Angle Theorem,
5.1 Angle Relationships in a Triangle
Chapter 4 Notes Classify triangles according to their sides
Triangles & Congruence Advanced Geometry Triangle Congruence Lesson 1.
Objectives: Use properties of isosceles and equilateral triangles
Isosceles, Equilateral, and Right Triangles Geometry Mrs. Kinser Fall 2012.
Warm Up Lesson Presentation Lesson Quiz Holt Geometry.
Chapter 4 Triangle Congruence By: Maya Richards 5 th Period Geometry.
Isosceles Triangle ABC Vertex Angle Leg Base Base Angles.
Chapter 4.1 Common Core - G.SRT.5 Use congruence…criteria for triangles to solve problems and prove relationships in geometric figures. Objectives – To.
Triangle Congruency Classifying Triangles by Sides Equilateral Triangle 3 congruent sides Isosceles Triangle At least 2 congruent sides Scalene Triangle.
Triangles : a three-sided polygon Polygon: a closed figure in a plane that is made of segments, called sides, that intersect only at their endpoints,
4.3 Isosceles & Equilateral Triangles Geometry Big Daddy Flynn 2013.
POINTS, LINES AND PLANES Learning Target 5D I can read and write two column proofs involving Triangle Congruence. Geometry 5-3, 5-5 & 5-6 Proving Triangles.
Chapter 9 Parallel Lines
4-1 Classifying Triangles I. Geometric Shapes What is a triangle? A TRIANGLE is a three-sided polygon.
Angles of a Triangle and Congruent Triangles April 24, 2008.
Proving Triangle Congruency. What does it mean for triangles to be congruent? Congruent triangles are identical meaning that their side lengths and angle.
Pythagorean Theorem Theorem. a² + b² = c² a b c p. 20.
Geometry - Unit 4 $100 Congruent Polygons Congruent Triangles Angle Measures Proofs $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400.
Bell Work 12/12 State which two triangles, if any, are congruent, and write a congruence statement and reason why 1) 2) Solve for the variables 3) 4)
4-4 Using Corresponding Parts of Congruent Triangles I can determine whether corresponding parts of triangles are congruent. I can write a two column proof.
Congruent Triangles Unit 4-5 Congruent Triangle Theorems.
Do-Now 2) Find the value of x & the measure of each angle. 5x – 4 4x ° 1) Find the value of x. 4x x – 10 3x + 1 5x – 4 + 4x + 14 = 100 9x.
Sect. 4.6 Isosceles, Equilateral, and Right Triangles
Chapters 2 – 4 Proofs practice. Chapter 2 Proofs Practice Commonly used properties, definitions, and postulates  Transitive property  Substitution property.
 Objective: we will be able to classify triangles by their angles and by their sides. A B C The vertices of a triangle are labeled with upper case letters.
Review: Solving Systems x 2y+3 x+y 12 Find the values of x and y that make the following triangles congruent.
Introduction Think of all the different kinds of triangles you can create. What are the similarities among the triangles? What are the differences? Are.
Geometry: Congruent Triangles
Proving Triangles Congruent
3.5 Parallel Lines and Triangles
Lines, Angles, and Triangles
The Isosceles Triangle Theorems
Congruent Triangles Unit 3.
Geometry Extra Credit Chapter 3
4.1 Congruent Figures -Congruent Polygons: have corresponding angles and sides -Theorem 4.1: If 2 angles of 1 triangle are congruent to 2 angles of another.
Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles.
Isosceles/ Equilateral
Lesson 4-6 Isosceles Triangles.
Presentation transcript:

C. N. Colon Geometry St. Barnabas HS

Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles triangles are a distinct classification of triangles with unique characteristics and parts that have specific names. In this lesson, we will explore the qualities of isosceles triangles.

Isosceles triangles have at least two congruent sides, called legs. The angle created by the intersection of the legs is called the vertex angle. Opposite the vertex angle is the base of the isosceles triangle. Each of the remaining angles is referred to as a base angle. The intersection of one leg and the base of the isosceles triangle creates a base angle. Key Concepts

Theorem Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the congruent sides are congruent. Key Concepts

If the Isosceles Triangle Theorem is reversed, then that statement is also true. This is known as the Converse of the Isosceles Triangle Theorem. Key Concepts

Theorem Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Key Concepts

If the vertex angle of an isosceles triangle is bisected, the bisector is perpendicular to the base, creating two right triangles. In the diagram that follows, D is the midpoint of. Key Concepts

Equilateral triangles are a special type of isosceles triangle, for which each side of the triangle is congruent. If all sides of a triangle are congruent, then all angles have the same measure. Key Concepts

Theorem If a triangle is equilateral then it is equiangular, or has equal angles. Key Concepts

Each angle of an equilateral triangle measures 60˚ Conversely, if a triangle has equal angles, it is equilateral. Key Concepts

Key Concepts, continued Theorem If a triangle is equiangular, then it is equilateral. Key Concepts

These theorems and properties can be used to solve many triangle problems. Key Concepts

Common Errors/Misconceptions incorrectly identifying parts of isosceles triangles not identifying equilateral triangles as having the same properties of isosceles triangles incorrectly setting up and solving equations to find unknown measures of triangles misidentifying or leaving out theorems, postulates, or definitions when writing proofs

YOU TRY Determine whether with vertices A (–4, 5), B (–1, –4), and C (5, 2) is an isosceles triangle. If it is isosceles, name a pair of congruent angles.

Use the distance formula to calculate the length of each side. Calculate the length of.

Substitute A (–4, 5) and B (–1, –4) for (x 1, y 1 ) and (x 2, y 2 ). Simplify. Calculate the length of

Substitute (–1, –4) and (5, 2) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

Calculate the length of. Substitute (–4, 5) and (5, 2) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

Determine if the triangle is isosceles. A triangle with at least two congruent sides is an isosceles triangle., so is isosceles.

Identify congruent angles. If two sides of a triangle are congruent, then the angles opposite the sides are congruent. ✔

Geogebra is a graphing program that can be used to illustrate the properties of isosceles triangles.

Find the values of x and y.

Make observations about the figure. The triangle in the diagram has three congruent sides. A triangle with three congruent sides is equilateral. Equilateral triangles are also equiangular.

The measure of each angle of an equilateral triangle is 60˚. An exterior angle is also included in the diagram. The measure of an exterior angle is the supplement of the adjacent interior angle.

Determine the value of x. The measure of each angle of an equilateral triangle is 60˚. Create and solve an equation for x using this information.

The value of x is 9. Equation Solve for x.

Determine the value of y. The exterior angle is the supplement to the interior angle. The interior angle is 60˚ by the properties of equilateral triangles. The sum of the measures of an exterior angle and interior angle pair equals 180. Create and solve an equation for y using this information.

The value of y is 13. Equation Simplify. Solve for y. ✔

Using Geogebra to solve this problem:

p. 361# 4-20 (mo4)

C. N. Colόn Geometry Reference: SIMON PEREZ.

CONGRUENT TRIANGLES CPCTC Corresponding Parts of Congruent Triangles are Congruent CPCTC ABC KLM by CPCTC BC A L M K We would have to prove that all six pairs of corresponding parts are congruent!

SSS ABC KLM by SSS BC A L M K We only had to prove that three pairs of corresponding parts are congruent! CONGRUENT TRIANGLES by Side-Side-Side

SAS ABC KLM by SAS BC A L M K We only had to prove that three pairs of corresponding parts are congruent! CONGRUENT TRIANGLES by Side-Angle-Side

ASA ABC KLM by ASA BC A L M K We only had to prove that three pairs of corresponding parts are congruent! CONGRUENT TRIANGLES by Angle-Side-Angle

AAS ABC KLM by AAS BC A L M K We only had to prove that three pairs of corresponding parts are congruent! CONGRUENT TRIANGLES by Angle-Angle-Side

SSS RST JKL by SSS SAS RST JKL by SAS ASA RST JKL by ASA AAS RST JKL by AAS SUMMARY: CONGRUENCE THEOREMS IN TRIANGLES R J S K T L R J S K T L R J S K T L R J S K T L

List the parts that are missing to be marked as congruent for both triangles to be congruent by AAS: AAS RST JKL by AAS S T R K L J If SRTKJL STR KLJ Missing part RS KJ

42 Missing parts CPCTC ABC KLM by CPCTC BC A L M K List the parts that are missing to be marked as congruent for both triangles to be congruent by CPCTC: IFTHEN AB KL AC KM BC LM BAC LKM ABC KLM ACB KML

List the parts that are missing to be marked as congruent for both triangles to be congruent by SAS: SAS RST JKL by SAS S T R K L J SR KJ RT JL If Missing part SRT KJL

List the parts that are missing to be marked as congruent for both triangles to be congruent by SSS: SSS RST JKL by SSS S T R K L J SR KJ RT JL If TS LK Missing part

List the parts that are missing to be marked as congruent for both triangles to be congruent by ASA: ASA RST JKL by ASA S T R K L J RT JL If SRT KJL Missing part STR KLJ

S T R K L J LL RST JKL by LL RIGHT TRIANGLES CONGRUENT by LEG - LEG We only had to prove that two pairs of corresponding parts are congruent!

S T R K L J L A RST JKL by L A RIGHT TRIANGLES CONGRUENT by LEG-ANGLE We only had to prove that two pairs of corresponding parts are congruent!

S T R K L J H L RST JKL by H L RIGHT TRIANGLES CONGRUENT BY HYPOTENUSE - LEG We only had to prove that two pairs of corresponding parts are congruent!

S T R K L J HA RST JKL by HA RIGHT TRIANGLES CONGRUENT by HYPOTENUSE - ANGLE We only had to prove that two pairs of corresponding parts are congruent!

HL RST JKL by HL LL RST JKL by LL LA RST JKL by LA HA RST JKL by HA SUMMARY: CONGRUENCE THEOREMS IN RIGHT TRIANGLES R J S K T L R J S K T L R J S K T L R J S K T L

S T R K L J List the parts that are missing to be marked as congruent for both triangles to be congruent by LA LA RST JKL by LA If STR KLJ Missing part RS KJ

p. 366 # 4, 12 and 14