Geometry Definitions, Postulates, and Theorems. Chapter 4: Congruent Triangles. Section 4.1: Apply Triangle Sum Properties

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1 Geometry efinitions, Postulates, and Theorems Key hapter 4: ongruent Triangles Section 4.1: pply Triangle Sum Properties Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. Triangle polygon formed by three segments joining three non-collinear points. triangle can be classified by its sides and then by its angles. *lassifying Triangles by Sides: Scalene Triangle triangle with no congruent sides. Leg ase ngle Isosceles Triangle triangle with at least two congruent sides. Vertex ngle Legs The congruent sides of an isosceles triangle, when only two sides are congruent. ase Leg ase ngle ase The third side (non-congruent side) of an isosceles triangle. quilateral Triangle triangle with three congruent sides. *lassifying Triangles by ngles: 45 cute Triangle triangle with three acute angles Leg Hypotenuse Right Triangle triangle with one right angle. Legs The sides that form the right angle. Hypotenuse The side opposite the right angle. Leg Obtuse Triangle triangle with one obtuse angle. 50 quiangular Triangle triangle with three congruent acute angles. (over)

2 x. lassify the triangles by their sides and angles. a) b) c) 120 Isosceles Obtuse quilateral quiangular/cute Scalene Right Vertex (plural: vertices) ach of the three points joining the sides of a triangle. djacent Sides of an ngle Two sides that share a common vertex. Opposite Side from an ngle The side that does not form the angle. Interior angles When the sides of a triangle are extended, additional angles are formed. The original angles are the interior angles. xterior angles When the sides of a triangle are extended, additional angles are formed. The angles that form linear pairs with the interior angles are the exterior angles. ***Theorem 4.1 Triangle Sum Theorem The sum of the measures of the three interior angles of a triangle is m = 67 0 ***Theorem 4.2 xterior ngle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles m 1 = orollary To The Triangle Sum Theorem The acute angles of a right triangle are complementary (add up to equal 90 ). Y X Z (over)

3 x. triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle. ( 5,4), (2,6), (4, 1) Yes, right triangle Isosceles x. ind the value of x and y. x. ind the value of x. Then classify the triangle by its angles. Then classify the triangle by its angles x Linear Pair x y (2x-18) x. ind the angle measures of the numbered angles x. ind the values of x and y. y 0 x

4 Section 4.2: pply ongruence and Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Two geometric figures are congruent if they have exactly the same size and shape, like placing one figure perfectly onto another figure. Y V ongruent igures ll the parts of one figure are congruent to the corresponding parts of the other figure. In congruent polygons, the corresponding sides and the corresponding angles are congruent. X W ongruence When writing a congruence statement for two polygons, always list the corresponding vertices in the same order! x. GIVN: orresponding ngles orresponding Sides x. Write a congruency statement. x. JKHL. ind the value of x and y. T L 9 cm J Q S x 3 cm R 86 0 (5y 12) 0 K P U H ***Theorem 4.3 Third ngle Theorem I two angles of one triangle are congruent to two angles of another triangle, THN, the third angles are also congruent. x. x G H K I J (over)

5 ***Theorem 4.4 Properties of ongruent Triangles Reflexive Property of ongruent Triangles or any triangle :. Symmetric Property of ongruent Triangles I, then. Transitive Property of ongruent Triangles I, and JKL, then JKL. x. ind the values of x and y. ( 8 y 0 0 x 2 ) ( 6x y)

6 Section 4.3: Prove Triangles ongruent by SSS Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Side-Side-Side (SSS) ongruence Postulate I three sides of one triangle are congruent to three sides of a second triangle, THN the two triangles are congruent. *Information about the angles is not needed. If Side, Side, and Side, then by. x. Is the congruence statement true? xplain your reasoning. WXY YZW x. Is the congruence statement true? xplain your reasoning. KJL MJL X K L Y W Z J M x. Write a proof. Given: Prove:, Reflexive (over)

7 Structural Support diagonal support added to a figure helps make the figure stable. The diagonal support forms triangles with fixed side lengths. y the SSS ongruence Postulate, these triangles cannot change shape and so the figure is stable. x. etermine whether the figure is stable. xplain your answer. a) b) c) Given:,, Given: Prove: W X Z Y WX YX Z is the midpoint of WY Prove: WXZ YXZ 1.,, 1. Given 1. WX YX 1. Given Z is the midpoint of WY 2. = = WZ ZY 3. + = 4. + = 4. Substitution = ef. of ongruent Segments SSS

8 Section 4.4: Prove Triangles ongruent by SS and HL Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Side-ngle-Side (SS) ongruence Postulate I two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, THN the two triangles are congruent. If Side, ngle, and Side, then by. x. o you have enough information to prove the triangles are congruent by SS? a) b) x. Write a proof. Given: Prove:, Vertical ngles are ongruent (over)

9 x. Write a proof. M Given: Prove: P, M P M MP P lines form rights angles 3. M MP Reflexive Property 5. M MP 5. Right Triangles: Legs The sides adjacent to the right angle. Hypotenuse The side opposite the right angle. ***Hypotenuse-Leg (HL) ongruence Theorem I the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, THN the two triangles are congruent. x. Write a proof. Given: Prove:, lines form rights angles Reflexive Property 5. 5.

10 Section 4.5: Prove Triangles ongruent by S and S Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***ngle-side-ngle (S) ongruence Postulate I two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, THN the two triangles are congruent. If ngle, Side, and ngle, then by. ***ngle-ngle-side (S) ongruence Postulate I two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, THN the two triangles are congruent. If ngle, ngle, and Side, then by. x. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem used. a) / b) / c) / d) / (over)

11 x. Write a proof. X Given: WZ bisects XZY and XWY Prove: WZX WZY Z W Y efinition of an angle bisector 3. ZW ZW WZX WZY 4. x. Write a proof. Given:,, M is the midpoint of Prove: M M M efinition of a midpoint 3. M M 3.

12 Section 4.6: Use ongruent Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***orresponding Parts of ongruent Triangles are ongruent (.P..T..) 1. Prove two triangles are congruent with SSS, SS, HL, S, or S. 2. Then, conclude that the corresponding parts of these congruent triangles are congruent as well. The triangles below are congruent by SS. Since the triangles are congruent, we know that: Y X Z x. Write a proof. H J Given: Prove: HJ II LK, JK II HL LHJ JKL L K lternate Interior ngles are ongruent 3. JL JL LHJ JKL LHJ JKL 5. x. Write a proof. M R Given: MS II TR, MS TR Prove: is the midpoint of MT S T lternate Interior ngles are ongruent 3. MS TR PT 5. is the midpoint of MT 5. (over)

13 x. Write a proof. P Given: MP bisects Prove: LP NP LMN, LM NM N L M 2. NMP LMP Reflexive Property 4. NMP LMP LP NP 5. x. Which triangles can you show are congruent in order to prove the statement? What postulate or theorem would you use? a) b) SW TY S W X Y T

14 Section 4.7: Use Isosceles and quilateral Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Vertex ngle Legs The two congruent sides in an isosceles triangle. Leg Leg Vertex ngle The angle formed by the legs in an isosceles triangle. ase The third side of the isosceles triangle. ase ngle ase ase ngles The two angles adjacent to the base in an isosceles triangle. ase ngle ***Theorem 4.7 ase ngles Theorem I two sides of a triangle are congruent, THN the angles opposite them are congruent. 30 x ***Theorem 4.8 onverse of ase ngles Theorem I two angles of a triangle are congruent, THN the sides opposite them are congruent. 2x-4 2x+2 x+5 orollary To ase ngles Theorem I a triangle is equilateral, THN it is equiangular. orollary to the onverse of ase ngles Theorem I a triangle is equiangular, THN it is equilateral.

15 Section 4.8: Perform ongruence Transformations Standards: 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. Transformation n operation that moves or changes a geometric figure in some way to produce a new figure. Image The new figure produced. Pre-Image Image ' *Three Types of Transformations: ' ' ' ' Translation Moves every point of a figure the same distance in the same direction. Translate down 6 Translate right 10 ' Reflection Uses a line of reflection to create a mirror image of the original figure. Rotation Turns a figure about a fixed point, called the center of rotation. Rotate 90 degrees clockwise ongruence Transformation hanges the position of the figure without changing its size or shape. Translate igure In The oordinate Plane Moving an object a given distance right or left and up or down. y *oordinate Notation for a Translation You can describe a translation by the notation ( x, y) ( x a, y b) which shows that each point ( x, y) of a figure is translated horizontally a units and vertically b units x. igure has the vertices ( 4,2), ( 2,5), ( 1,1), and ( 3, 1). Sketch and its image after the translation ( x, y) ( x 5, y 2). Right 5 own 2 ' ' ' ' x (over)

16 Usually Reflect igure In The oordinate Plane The line of reflection is always the x-axis or the y-axis. y *oordinate Notation for a Reflection Reflection in the x-axis: ( x, y) ( x, y) Multiply the y-coordinate by -1. Reflection in the y-axis: ( x, y) ( x, y) Multiply the x-coordinate by -1. x x. Use a reflection in the x-axis to draw the other half of the figure. Rotate igure In The oordinate Plane The center of rotation is the origin. The direction of rotation can be either clockwise or counterclockwise. The angle of rotation is formed by rays drawn from the center of rotation through corresponding points on the original figure and its image clockwise rotation 60 0 counterclockwise rotation x. Graph PQ and RS. Tell whether RS is a rotation of PQ about the origin. If so, give the angle and direction of rotation. y P a) P ( 2,6), Q(5,1), R(6, 1), S(1, 2) Q x b) P ( 4,2), Q(3,3), R( 2,4), S( 3,3) S R

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