CHAPTER # 4 CONGRUENT TRIANGLES

Transcription

1 HPTER # 4 ONGRUENT TRINGLES In this chapter we address three ig IES: 1) lassify triangles by sides and angles 2) Prove that triangles are congruent 3) Use coordinate geometry to investigate triangle relationships Section: Essential Question 4 1 pply Triangle Sum Properties How can you find the measure of the third angle of a triangle if you know the measures of the other two angles? Warm Up: Key Vocab: Triangle a polygon with three sides Scalene Triangle a triangle with NO congruent sides Isosceles Triangle a triangle with T LEST two congruent sides Equilateral Triangle a triangle with three congruent sides Student Notes Geometry hapter 4 ongruent Triangles KEY Page #1

2 cute Triangle a triangle with three acute angles Right Triangle a triangle with one right angle Obtuse Triangle a triangle with one obtuse angle Equiangular Triangle a triangle with three congruent angles Interior ngle Exterior ngle When the sides of a polygon are extended, the interior angles are the original angles. When the sides of a polygon are extended, the exterior angles are the angles that form linear pairs with the interior angles. Interior angles Exterior angles orollary to a Theorem statement that can be proved easily using the theorem to which it is linked. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #2

3 Theorems: The sum of the measures of a triangle is 180 Triangle Sum Theorem m m m 180 The acute angles of a right triangle are complementary orollary to the Triangle Sum Theorem m m 90 Exterior ngle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. m 1 m m 1 Show: Ex 1: lassify by its sides and by its angles. Sides: 2 2 (2 5) (6 4) (4 2) ( 1 6) (4 5) ( 1 4) ngles: m m m y x ecause N, is an Isosceles Right Triangle Student Notes Geometry hapter 4 ongruent Triangles KEY Page #3

4 Ex 2: ind m E. y the Exterior ngle Theorem: 3x 6 80 x 2x 74 x 37 3(37) (3x+6) x G E Ex 3: The support for the skateboard ramp shown forms a right triangle. The measure of one acute angle in the triangles is five times the measure of the other. ind the measure of each acute angle. y the orollary to the Triangle Sum Theorem: x 5x 90 6x 90 x 15 x 15 5x 75 Student Notes Geometry hapter 4 ongruent Triangles KEY Page #4

5 Section: Essential Question 4 2 pply ongruence and Triangles What are congruent figures? Warm Up: Key Vocab: ongruent igures orresponding Parts Two or more figures with exactly the same size and shape. ll corresponding parts, sides and angle, are congruent. pair of sides or angles that have the same relative position in two or more congruent figures Theorems: Third ngles Theorem If Then two angles of one triangle are congruent to the third angles are also congruent. two angles of another triangle, and E,. E E Student Notes Geometry hapter 4 ongruent Triangles KEY Page #5

6 Properties: Reflexive ongruence of Triangles Triangle congruence is reflexive, symmetric, and transitive. Symmetric If E, then E Transitive If E and E JKL, then JKL Show: Ex 1: Write a congruence statement for the triangles shown. Identify all pairs of congruent corresponding parts NMO YXZ N M O Y X Z NO YZ; NM YX ; MO XZ MNO XYZ ; OMN ZXY ; MON XZY Ex 2: In the diagram, (3x-6) in GHK G in 80 H 44 K a. ind the value of x. 3x 6 9 3x 15 x 5 (4x-2y) b. ind the value of y. 4x 2y 44 4(5) 2y 44 2y 24 y 12 Ex 3: ind m YXW. W X Y Z m YXW Student Notes Geometry hapter 4 ongruent Triangles KEY Page #6

7 S T Ex 4: Given: SV RV, TV WV, ST RW, T W Prove: STV RWV V W R Statements Reasons 1. SV RV, TV WV, ST RW, T W 1. Given 2. SVT RVW 2. Vert. ' s Thm 3. S R 3. Third ' s Thm 4. STV RWV 4. ef. of 's L M Ex 5: Given: Quad LMNO is a square MO bisects LMN and NOL Prove: OLM ONM O N 1. Statements Quad LMNO is a square MO bisects LMN and NOL 1. Given Reasons 2. OL ON NM ML ; L N 3. LOM MON LMO OMN 2. efinition of a Square 3. efinition of an angle bisector 4. OM OM 4. Reflexive Prop. 5. OLM ONM 5. ef. of 's losure: How do you know two figures are congruent? LL corresponding sides and LL corresponding angles must be congruent. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #7

8 Section: Essential Question 4 3, 4 9 ongruence Transformations How do you identify a rigid motion in the plane? What transformations create an image congruent to the original figure? Warm Up: Key Vocab: Transformation Image Rigid Motion Translation Reflection Rotation n operation that moves or changes a geometric figure in some way to produce a new figure. The new figure that is produced in a transformation transformation that preserves length, angle measure, and area. lso called isometry. transformation that moves every point of a figure the same distance in the same direction. oordinate Notation: ( x k, y k) transformation that uses a line of reflection to create a mirror image of the original figure. oordinate Notation: x-axis: ( x, y) y-axis: ( xy, ) transformation in which a figure is turned about a fixed point called the center of rotation. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #8

9 Show: Ex 1: Name the type of transformation demonstrated in each picture. a. b. c. Translation in a straight path Rotation about a point Reflection in a vertical line Ex 2: escribe the transformation(s) you can use to move figure onto figure. Rotation about P and then a reflection P Ex 3: igure WXYZ has the vertices W ( 1,2), X (2,3), Y(5,0), and Z(1, 1). Sketch WXYZ and its image after the translation ( xy, ) ( x 1, y 3). W y X Y x -1-2 Z Student Notes Geometry hapter 4 ongruent Triangles KEY Page #9

10 Ex 4: Reflect the point across the x-axis, the y-axis, and then rotate it 180 through the origin. Record the coordinates of each of these points below. x-axis: (5, 3) y-axis: ( 5, 3) 180 rotation: ( 5, 3) Student Notes Geometry hapter 4 ongruent Triangles KEY Page #10

11 Section: Essential Question 4 4 Prove Triangles ongruent by SSS How can you use side lengths to prove triangles congruent? Warm Up: Postulate: If three sides of one triangle are congruent to three sides of a second triangle, Side-side-side (SSS) ongruence Postulate then the two triangles are congruent. E, E, and E E Show: Ex1: Which are the coordinates of the vertices of a triangle congruent to XYZ? y. (6,2),(0, 6),(6, 5) (5,1),( 1, 6),(5, 6) Y (4,0),( 1, 7),(4, 7) x. (3, 1),( 3, 7),(3, 8) X Z -7 Student Notes Geometry hapter 4 ongruent Triangles KEY Page #11

12 Ex2: Given: iagram Prove Statements Reasons 1. ; 1. Given Reflexive Prop SSS Post. Try: Ex3: Given: is the midpoint of Prove: Statements Reasons 1. is the midpoint of ; a. Given Reflexive Property efinition of a midpoint SSS Postulate losure: an you use side lengths to prove quadrilaterals congruent? No, SSS can only be applied to triangles. our sides can be arranged in different orders to create different quadrilaterals, whereas three sides will create a unique triangle. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #12

13 Leg Section: Essential Question 4 5 Prove Triangles ongruent by SS and HL How can you use two sides and an angle to prove triangles congruent? Warm Up: Key Vocab: Legs (of a Right Triangle) Hypotenuse In a right triangle, the sides adjacent to the right angle. In a right triangle, the side opposite the right angle lways the longest side of a right triangle Hypotenuse Leg Side-ngle-Side (SS) ongruence Postulate If then two sides and the included angle of one triangle are congruent to two sides and the the two triangles are congruent. included angle of a second triangle, E,, and E E Student Notes Geometry hapter 4 ongruent Triangles KEY Page #13

14 Hypotenuse-Leg (HL) Theorem If then the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a the two triangles are congruent. second right triangle, E, E, and and E are right triangles E E Show: Ex 1: If you know that and, what postulate or theorem can you use to conclude that? The SS Post. Ex 2: In the diagram R is the center of the circle. If SRT URT, what can you conclude about SRT and URT They are congruent by SS Post. R S T U Ex 3: State the third congruence that would allow you to prove RST XYZ by the SS ongruence Postulate. a. ST YZ, RS XY S Y b. T Z, RT XZ ST YZ Student Notes Geometry hapter 4 ongruent Triangles KEY Page #14

15 X Ex 4: Given: YW XZ; XY ZY Prove: XYW ZYW Y W Z Statements 1. YW XZ; XY ZY 1. Given Reasons 2. XWY and ZWY are rt. ' s 2. lines form 4 rt. 's 3. XYW ZYW are rt. ' s 3. ef. of rt. 4. YW YW 4. Reflexive Prop. 5. XYW ZYW 5. HL Thm. M N Ex 5: Given: MP NP; OP bisects MPN Prove: MOP NOP O P Statements Reasons 1. MP NP; OP bisects MPN 1. Given 2. MPO NPO 2. ef. of bis. 3. OP OP 3. Reflexive Prop 4. MOP NOP 4. SS Post. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #15

16 Section: Essential Question 4 6 Prove Triangles ongruent by S and S If one side of a triangle is congruent to one side of another, what do you need to know about the angles to prove the triangles are congruent? Warm Up: Postulates: ngle-side-ngle (S) ongruence Postulate If then two angles and the included side of one triangle are congruent to two angles and the the two triangles are congruent. included side of a second triangle,, E, and E E E Student Notes Geometry hapter 4 ongruent Triangles KEY Page #16

17 Theorems: ngle-ngle-side (S) ongruence Theorem If Then two angles and a non-included side of one triangle are congruent to two angles and a the two triangles are congruent. non-included side of a second triangle, E,, and E E Show: Ex 1: an the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. a. b. c. S Post S Theorem annot be proven congruent Ex 2: Write a two-column proof. Given: ; E E ; Prove: E E Statements Reasons 1. ; E E 1. Given 2. is a rt. ; E is a rt. 2. ef. of lines 3. E 3. Rt. Thm. 4. ; 4. Given 5. E 5. S Post. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #17

18 Key Vocab: low Proof Type of proof that uses arrows to show the flow of a logical argument. Ex 3: Write a flow proof for Example 2. ; E E (Given) ; (Given) is a rt. ; E is a rt. (ef. of lines) E (S Post.) E (Rt. 's Thm) Ex 4: Write a flow proof: Given: Prove: E (Given) E ( Supps. Thm.), are supplements;, E are supplements (Linear Pair Post.) E E (S Post.) (Given) E (Vert. 's Thm.) losure: What are the IVE ways to prove that two triangles are congruent? 1. SSS Postulate 2. SS Postulate 3. HL Theorem 4. S Postulate 5. S Theorem Student Notes Geometry hapter 4 ongruent Triangles KEY Page #18

19 Section: Essential Question 4 7 Use ongruent Triangles How can you use congruent triangles to prove angles or sides congruent? Warm Up: Key Vocab: PT orresponding Parts of ongruent Triangles are ongruent Show: Ex 1: If P is the midpoint of MS, how wide is the bull s pasture? 40 feet M N P 30 ft R 40 ft S Student Notes Geometry hapter 4 ongruent Triangles KEY Page #19

20 Ex 2: Write a two-column proof. Given: GK bisects GH and KH Prove: K HK K G Statements 1. GK bisects GH and KH 1. Given 2. GK HGK; KG HKG 2. ef. of bis. 3. GK GK 3. Reflexive Prop. 4. GK HGK 4. S Post. 5. K HK 5. PT H Reasons O Ex 3: Write a flow proof: N 3 4 P 3 4 (Given) Given: 1 2; 3 4 Prove: MNR QPR 3 and 5 are suppl (Linear Pair Post) 4 and 6 are suppl (Linear Pair Post) M 1 5 R 6 2 Q 5 6 (ongruent Suppl s Thm) 1 2 (Given) MQ MQ (Reflexive Prop.) MNQ QPM (S ) MN PQ (PT) MRN QRP (Vert. ng s ong. Thm) losure: MNR QPR (S ) How can you use congruent triangles to prove angles or sides are congruent? PT- If two triangles are congruent, then all of their corresponding parts are also congruent. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #20

21 Section: Essential Question 4 8 Use Isosceles and Equilateral Triangles How are the sides and angles of a triangle related if there are two or more congruent sides or angles? Warm Up: Key Vocab: Legs omponents of an Isosceles Triangle The congruent sides vertex Vertex ngle The angle formed by the legs legs ase The third side (the side that is NOT a leg) ase ngle The two angles that are adjacent to the base base base angles Theorems: If ase ngles Theorem (Isosceles Triangle Theorem) then two sides of a triangle are congruent, the angles opposite them are congruent. Student Notes Geometry hapter 4 ongruent Triangles KEY Page #21

22 ase ngles Theorem onverse (Isosceles Triangle Theorem onverse) If then two angles of a triangle are congruent, the sides opposite them are congruent. orollaries: If a triangle is equilateral, then it is equiangular. If it is equiangular. then a triangle is equilateral, Show: Ex 1: In PQR, PQ PR. Name two congruent angles. Q Q R P Ex 2: ind the measure of R X and Z. Y 50 65,65 X Z Student Notes Geometry hapter 4 ongruent Triangles KEY Page #22

23 Ex 3: ind the values of x and y in the diagram. 7 x 7, y 3 x y+4 Ex 4: iagonal braces and are used to reinforce a signboard that advertises fresh eggs and produce at a roadside stand. Each brace is 14 feet long. a. What congruent postulate can you use to prove that? 8 ft 8 ft E SSS Post. b. Explain why E is isosceles., since PT. E E by the onv. of the ase 's Thm. nd this implies that E is isosceles. c. What triangles would you use to show that E is isosceles? and Student Notes Geometry hapter 4 ongruent Triangles KEY Page #23