Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION Standards 4 and 5 REFLEXIVE,

Published byPeter Lucas Modified over 9 years ago

Similar presentations

Presentation on theme: "1 PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION Standards 4 and 5 REFLEXIVE,"— Presentation transcript:

1

2 1 PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION Standards 4 and 5 REFLEXIVE, SYMMETRIC AND TRANSITIVE PROPERTIES EXTERIOR ANGLE THEOREM CPCTC ANGLE SUM THEOREM PROBLEM 5APROBLEM 5B END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 2 STANDARD 4: Students prove basic theorems involving congruence and similarity. ESTÁNDAR 4: Los estudiantes prueban teoremas que involucran congruencia y semejanza. STANDARD 5: Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ESTÁNDAR 5: Los estudiantes prueban que son triángulos congruentes o semejantes y son capaces de usar el concepto de partes correspondientes de triángulos congruentes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 3 These are examples of POLYGONS: These are NOT POLYGONS: A POLYGON is a closed figure in a plane which is made up of line segments, called sides, that intersect only at their endpoints, named vertices. Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 4 A TRIANGLE is a three-sided polygon Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 5 55° 64°61° 21° 110° 49° RIGHT TRIANGLE ACUTE TRIANGLE OBTUSE TRIANGLE ANGLES CLASSIFYING TRIANGLES BY ANGLES STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 6 Parts of a RIGHT TRIANGLE Leg HYPOTENUSE Right Angle PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 SIDES CLASSIFYING TRIANGLES BY SIDES 10 4 9 22 13 13 13 SCALENE TRIANGLE ISOSCELES TRIANGLE EQUILATERAL TRIANGLE 9 Standards 4 and 5 Also EQUIANGULAR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 8 Parts of an ISOSCELES TRIANGLE Base Angles Base Leg Vertex Angle Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 9 B C A REFLEXIVE CONGRUENCE of triangles is REFLEXIVE: ABC ABC Standards 4 and 5 b c a PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 10 SYMMETRIC: CONGRUENCE of triangles is SYMMETRIC: A CB M K L IFTHEN ABC KLM KLM ABC Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 11 TRANSITIVE CONGRUENCE of triangles is TRANSITIVE A CBM K L IFAND R TS THEN ABC RST KLM RST ABC KLM Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 12 1 2 3 The measure of an exterior angle (angle 3) of a triangle is equal to the sum of the measures of the two remote interior angles (angles 1 and 2). + 1 m 2 m = 3 m Exterior Angle Theorem Remote Interior Angles Exterior Angle Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 13 Standards 4 and 5 (3X+4)° (7X+5)° (6X+69)° Z Find in the figure below: Z m PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 14 Standards 4 and 5 (3X+4)° (7X+5)° (6X+69)° Z (7X+5)°+(3X+4)° = (6X+69)° By Exterior Angle Theorem: 7X+5 + 3X+4 = 6X +69 10X + 9 = 6X +69 -9 10X = 6X + 60 -6X 4X = 60 4 X=15 By Linear Pair: Z m + (6X+69)° = 180° Z m + 6X+69 = 180 Z m + 6( )+69 = 180 15 Z m + 90+69 = 180 Z m + 159 = 180 -159 Z m = 21° Find in the figure below: Z m Substituting the value for X: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 15 Standards 4 and 5 (3Z+8)° (9Z+6)° (5Z+84)° X Find in the figure below: X m PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 16 Standards 4 and 5 (3Z+8)° (9Z+6)° (5Z+84)° X (9Z+6)°+(3Z+8)° = (5Z+84)° By Exterior Angle Theorem: 9Z+6 + 3Z+8 = 5Z +84 12Z + 14 = 5Z +84 -14 12Z = 5Z + 70 -5Z 7Z = 70 7 Z=10 By Linear Pair: X m + (5Z+84)° = 180° X m + 5Z + 84 = 180 X m + 5( )+84 = 180 10 X m + 50+84 = 180 X m + 134 = 180 -134 X m = 46° Find in the figure below: X m Substituting the value for Z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 17 CONGRUENT TRIANGLES CPCTC Corresponding Parts of Congruent Triangles are Congruent CPCTC ABC KLM by CPCTC BC A L M K Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 18 Standards 4 and 5 If RST UVW with RS = 6X + 8, UV = 56, RT = 2X – 2, What is the value of X, and RT. R T S U W V 6X+8 56 2X – 2 6X + 8 = 56 Since both triangles are congruent, corresponding sides are congruent as well: -8 6X = 48 6 X = 8 Now finding RT: RT=2X – 2 =2( ) – 2 8 = 16 – 2 RT= 14 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 19 Standards 4 and 5 If DEF JKL with DE = 5X + 3, JK = 48, DF = 3X + 1, What is the value of X, and DF. D F E J L K 5X+3 48 3X + 1 5X + 3 = 48 Since both triangles are congruent, corresponding sides are congruent as well: -3 5X = 45 5 X = 9 Now finding DF: DF=3X + 1 =3( ) + 1 9 = 27 + 1 DF= 28 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 20 Standards 4 and 5 Given that LMN RST with sides LM=5X+3, RS=3X+13, and ST = 6X + 9. Find the value of X and ST. L N M R T S 5X+3 3X+13 5X +3 = 3X + 13 Since both triangles are congruent, corresponding sides are congruent as well: -3 5X =3X + 10 2 Now finding ST: ST=6X + 9 =6( ) +9 5 = 30 + 9 6X + 9 -3X 2X = 10 X = 5 ST= 39 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 21 Standards 4 and 5 Given that EFG HIJ with sides EF =9X+2, HI=4X+17, and JI = 2X + 3. Find the value of X and JI. E G F H J I 9X+2 4X+17 9X +2 = 4X + 17 Since both triangles are congruent, corresponding sides are congruent as well: -2 9X =4X + 15 5 Now finding JI: JI=2X + 3 =2( ) +3 3 = 6 + 3 2X + 3 -4X 5X = 15 X = 3 JI= 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 22 35° 58° 87° ++ C C m B B m A A m = 180° 35° + 87° + 58° = 180° The sum of the interior angles of a triangle is always 180° ANGLE SUM THEOREM: Standards 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24 23 A A m = 180° B B m + C C m + If and A = 90 ° m then = 180° B m + C m + 90° - 90° -90° = 90° B m C m + Conclusion: In a right triangle, both non- right angles are acute and complementary! What kind of angles are the non-right angles in a right triangle? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25 24 Standards 4 and 5 If QRT UVW, with Q m = 100°, V m = 30°, and, find the value for Z. W m = 4Z +10 Q T RW V U Since both triangles are congruent then their corresponding angles are congruent: 4Z + 10 100° 30° Q = m U m Q U 100° = 100° Now from the Angle Sum Theorem: V m W m + =180° U m + 100° + 30° + = 180° W m 130 + = 180° W m -130° W = 50° m W m = 4Z +10 If then 4Z + 10 = 50 -10 100° + 30° + = 180° W m 4Z = 40 4 Z = 10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26 25 Standards 4 and 5 If DEF JKL, with D m = 110°, K m = 20°, and, find the value for Z. L m = 3Z +23 D F EL K J Since both triangles are congruent then their corresponding angles are congruent: 3Z + 23 110° 20° D = m J m D J 110° = 110° Now from the Angle Sum Theorem: K m L m + =180° J m + 110° + 20° + = 180° L m 130 + = 180° L m -130° L = 50° m L m = 3Z + 23 If then 3Z + 23 = 50 -23 110° + 20° + = 180° L m 3Z = 27 3 Z = 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

27 26 Standards 4 and 5 If FGH IJK with, and GFH= m 3X+28 JKI= m 2X – 8 KJI= m 2X+20, what is the value for x and for each angle? F H GK J I 3X+28 2X – 8 2X + 20° 3X + 28 Since both triangles are congruent then their corresponding angles are congruent: HFG KIJ HFG= m KIJ= m 3X+28 JKI + m KIJ+ m KJI= m 180° (2X – 8 ) + (3X + 28) + (2X+20) = 180 Now from the Angle Sum Theorem in KIJ: 2X – 8 + 3X + 28 + 2X+20 = 180 2X + 3X + 2X – 8 + 28+20 = 180 7X + 40 = 180 -40 7X = 140 7 JKI = m 2X – 8 = 2( ) – 8 = 40 - 8 = 32° 20 KIJ = m 3X+28 = 3( ) +28 = 60+28 = 88° 20 KJI = m 2X+20 = 2( ) +20 = 40+20 = 60° 20 X = 20 Checking solution: 32° + 88° +60° = 180° 180°=180° PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

28 27 Standards 4 and 5 If RST NOP with, and TRS= m 4X+20 NPO= m 5X + 6 PON= m 3X+10, what is the value for x and for each angle? R T SP O N 4X+20 5X +6 3X + 10° 4X + 20 Since both triangles are congruent then their corresponding angles are congruent: TRS PNO TRS = m PNO= m 4X+20 NPO+ m PNO+ m PON= m 180° (5X + 6 ) + (4X + 20) + (3X+10) = 180 Now from the Angle Sum Theorem in NOP: 5X + 6 + 4X + 20 + 3X+10 = 180 5X + 4X + 3X + 6 + 20+10 = 180 12X + 36 = 180 -36 12X = 144 12 NPO= m 5X + 6 = 5( ) +6 =60 + 6 = 66° 12 PNO= m 4X+20 = 4( ) +20 = 48+20 = 68° 12 PON= m 3X+10 = 3( ) +10 = 36+10 = 46° 12 X = 12 Checking solution: 66° + 68° +46° = 180° 180°=180° PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Download ppt "1 PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION Standards 4 and 5 REFLEXIVE,"

Similar presentations

PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION PRESENTATION CREATED BY.

PROBLEM 1A PROBLEM 2A PROBLEM 3A PROBLEM 4A PROBLEM 1B PROBLEM 4B PROBLEM 2B PROBLEM 3B TRIANGLES AS POLYGONS: CLASSIFICATION PRESENTATION CREATED BY.
Chapter 4a: Congruent Triangles By: Nate Hungate, Gary Russell, J. P

Chapter 4a: Congruent Triangles By: Nate Hungate, Gary Russell, J. P
PROPORTIONS SSS SIMILARITY PARALLEL TRANSVERSALS PROBLEM 1 PROBLEM 2

PROPORTIONS SSS SIMILARITY PARALLEL TRANSVERSALS PROBLEM 1 PROBLEM 2
1 The sum of 2 sides of the triangle greater than the other side? Ordering the angles of a triangle? Ordering the sides of a triangle? SAS Inequality.

1 The sum of 2 sides of the triangle greater than the other side? Ordering the angles of a triangle? Ordering the sides of a triangle? SAS Inequality.
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:

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:
Congruent Triangles Geometry Chapter 4.

Congruent Triangles Geometry Chapter 4.
Identifying Congruent Triangles

Identifying Congruent Triangles
TRIANGLE SUM PROPERTIES 4.1. TO CLARIFY******* A triangle is a polygon with three sides. A triangle with vertices A, B, and C is called triangle ABC.

TRIANGLE SUM PROPERTIES 4.1. TO CLARIFY******* A triangle is a polygon with three sides. A triangle with vertices A, B, and C is called triangle ABC.
4.1 Triangles and Angles.

4.1 Triangles and Angles.
ANGLES FORM BY A TRANSVERSAL PARALLEL LINES CUT BY A TRANSVERSAL

ANGLES FORM BY A TRANSVERSAL PARALLEL LINES CUT BY A TRANSVERSAL
Review: Begin at the word “We”. Every Time you move, write down the word(s) upon which you land. We in seven days! break another have will 1. Move to the.

Review: Begin at the word “We”. Every Time you move, write down the word(s) upon which you land. We in seven days! break another have will 1. Move to the.
C. N. Colon Geometry St. Barnabas HS. Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details,

C. N. Colon Geometry St. Barnabas HS. Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details,
TRIANGLES (There are three sides to every story!).

TRIANGLES (There are three sides to every story!).
Chapter 4 Congruent Triangles. 4.1 & 4.6 Triangles and Angles Triangle: a figure formed by three segments joining three noncollinear points. Classification.

Chapter 4 Congruent Triangles. 4.1 & 4.6 Triangles and Angles Triangle: a figure formed by three segments joining three noncollinear points. Classification.
Lesson 2.1 Classifying Triangles Triangle- a closed figure in a plane that is the union of three segments endpoint to endpoint. Interior Angle- an angle.

Lesson 2.1 Classifying Triangles Triangle- a closed figure in a plane that is the union of three segments endpoint to endpoint. Interior Angle- an angle.
4.1 Triangles & Angles August 15, 2013.

4.1 Triangles & Angles August 15, 2013.
Chapter 4 Congruent Triangles In this chapter, you will: classify triangles by their parts, apply the Angle Sum Theorem and the Exterior Angle Theorem,

Chapter 4 Congruent Triangles In this chapter, you will: classify triangles by their parts, apply the Angle Sum Theorem and the Exterior Angle Theorem,
3.4 & 4.5 Triangles.

3.4 & 4.5 Triangles.
Geometry. Kinds of triangles Geometry Kinds of triangles.

Geometry. Kinds of triangles Geometry Kinds of triangles.
4.1 Triangles and Angles Geometry Mrs. Spitz Fall 2004.

4.1 Triangles and Angles Geometry Mrs. Spitz Fall 2004.

Similar presentations

Ads by Google